Method of manufacturing light beam scanning apparatus and fixed hologram plate and rotatable hologram and light distributing apparatus

ABSTRACT

A high-resolution light-beam scanning apparatus utilizing only mass-producible holograms instead of utilizing auxiliary optical systems, and capable of compensating for disadvantages. The light-beam scanning apparatus including diffraction gratings for minimizing either: a sum total of values obtained by weighting a square of an optical path length difference between an optical path of a light flux measured along a principal axis of a light beam incident on and diffracted by a first diffraction grating of a rotatable hologram, and incident on and diffracted by a second diffraction grating of a fixed plate to conduct a scanning and converging on a scanning point on an image formation surface, and an optical path of a light flux measured along a marginal ray distanced from the principal axis or an absolute value of the optical path difference thereof; or a sum total of values obtained by weighting a square of a sum obtained by adding an amount of displacement of a light-beam convergent on a scanning point on the image formation surface, to an amount of displacement of the same light. The displacement measured with respect to the principal axis of a phase recorded on the diffraction grating when the light flux is incident on the fixed plate or by weighting an absolute value of the sum. The weighting is conducted at every scanning position of an image formation surface.

This application is a division of application Ser. No. 07/949,520, filedas PCT/JP92/00371, Mar. 26, 1992, now U.S. Pat. No. 5,680,253.

FIELD OF THE INVENTION

The present inventions pertain to a method of manufacturing a light beamscanning apparatus and a fixed hologram plate and to a rotatablehologram and a light distributing apparatus. More particularly, thepresent invention refers to a method of manufacturing a light scanningapparatus employing a hologram disk, a method of manufacturing fixedhologram plate, and to a rotatable hologram and a light distributingapparatus.

BACKGROUND ART

A high-precision and high-resolution laser-scanning optical system isused in office automation equipment including a laser printer and alaser facsimile, and in such apparatuses as a laser drawing apparatusand a laser inspection apparatus. Conventionally, this optical system isembodied by a rotating polygonal mirror and a combination of a pluralityof f-θ lenses.

In the above method employing a polygonal mirror, efforts to lower costhave met with difficulty because of the high precision required tofabricate a rotating polygonal mirror and because of a large number oflens groups required, including f-θ lenses that serve, at the same time,as inclination correction optical system.

On the other hand, a hologram scanning apparatus employing a hologramcan be mass produced. As an example of such a hologram scanningapparatus, the present applicant has filed an application for a hologramscanning apparatus for performing a scanning with a straight beam havinga high resolution and having sufficiently corrected aberration (theJapanese Laid-Open Patent Application 63-072633 and the JapaneseLaid-Open Patent Application 61-060846). This light beam scanningapparatus achieves, as a scanning optical system for a laser printer,excellent specifications characterized by a high precision, ensuring astable print quality. However, there is now a demand for alaser-scanning optical system having even higher resolution, on theorder of 400-600 dpi or even 1000 dpi. Also, further cost reduction isdesired.

In order to embody a hologram scanner having such an extremely highresolution at a low price, the following objectives need be resolved:

1 scanning beam radius should be as thin as 60 μm (equivalent to 400dpi), for example, and as uniform as possible; and

2 a scanning should be carried out at the same velocity as that of therotation of a rotatable hologram, which rotation is at constant angularacceleration.

Since a wavelength of a semiconductor laser used therein as a scanninglight source can vary according to ambient temperature and since severallongitudinal modes can be produced,

3 displacement in a scanning direction of a scanning beam should becompensated for; and

4 displacement in a cross scanning direction of a scanning beam shouldbe compensated for.

Since a scanning beam displacement is attributable to a warping of abase used in a rotatable hologram and the warping takes place as aresult of using a floating glass, which is of low cost and needs nopolishing, or a plastic base (PMMA, for example) enabling injectionmolding,

5 a scanning beam displacement due to the plastic base being moved fromits ideal position should be compensated for.

The present applicant had proposed a method of achieving the above tasksin the Japanese Laid-Open Patent Application 58-119098. The device usedin the method comprises, as shown in FIG. 14, a rotatable hologram 10and a fixed hologram plate 20 disposed between the rotatable hologram 10and an image formation surface 4. The hologram 10 is a rapidly rotatingrotatable hologram in which a plurality of hologram plates are disposed.Further, 5 is a reconstructing beam, 6 is a diffracted wave outgoingfrom the hologram plate 10, and 7 is a diffracted wave outgoing from thefixed hologram plate 20. The reconstructing beam from a semiconductornot shown in the figure is diagonally incident on the rotating rotatablehologram 10, whose rotation enables the scanning by the diffracted wave6. The diffracted wave 6 is incident on the fixed hologram plate 20, andthe diffracted wave 7, which is a wave diffracted therefrom, scans theimage formation surface 4.

In the above configuration, displacement of a scanning beam position dueto a wavelength variation of the semiconductor laser is compensated for,and a velocity of the scanning beam is maintained constant by a rotationof constant angular acceleration of the rotatable hologram 10, so that astraight-line scanning by a scanning beam is achieved. Further,displacement of a scanning beam position both in the scanning directionand the cross scanning direction, which displacement is due to awavelength variation of the semiconductor laser, is corrected by havingthe fixed hologram plate 20 bend the scanning beam in a directioncounter to a scanning direction of the rotatable hologram 10.

As an improved method of compensating for displacement of the scanningbeam position in the cross direction due to a wavelength variation ofthe semiconductor laser, the present applicant filed an application forthe Japanese Laid-Open Patent Application 60-168830, in which it isproposed that a fixed hologram plate be spatially placed before therotatable hologram.

The present applicant also made a proposition in the Japanese Laid-OpenPatent Application 2-179437 (the domestic declaration of priority on theJapanese Laid-Open Patent Application 1-240720), in which is proposed aconstruction capable, by employing at least two holograms, ofmaintaining uniform optical path lengths from an incident wave to animage formation surface, and of preventing degradation of wavefrontcharacteristics on the image formation surface, which degradation iscaused by a wavelength variation of the reconstructing light source.Since the Japanese Laid-Open Patent Application 2-179437 relates to anoptical system where at least two holograms, as mentioned above, arefixed, and therefore only one image formation point is provided, anapplication of the same device to the scanning optical system now beingdiscussed entails some difficulty in that moment-by-moment optical pathlength changes, which take place as the beam scanning proceeds,inevitably cause the optical path length to be longer at the scanningend than at the scanning center. Accordingly, the aforementionedconventional technology has not resolved all of the objectives from 1through 5 described earlier.

SUMMARY OF THE INVENTION

It is an object of the present invention to provide, as a solution tothe objectives 1 through 5 above a method of manufacturing ahigh-resolution light-beam scanning apparatus employing onlymass-producible holograms without using any auxiliary optical systemconsisting of optical lenses or mirrors having curvature. It is anotherobject of the present invention to provide a method of manufacturing afixed hologram plate.

Another object of the present invention is to provide a method ofmanufacturing a light beam scanning apparatus employing at least twoholograms, wherein a quality of a scanning beam or a scanningperformance does not show deterioration even when a wavelengthdisplacement owing to a wavelength variation or a wavelength dispersionof the light source takes place.

Yet a further object of the present invention is to provide a method ofmanufacturing a hologram plate.

A further object of the present invention is to provide a light-beamscanning apparatus employing at least two holograms, wherein coloraberration due to a wavelength displacement caused by a wavelengthvariation or wavelength dispersion of the light source is corrected.

A still further object of the present invention is to obtain a hologramconstruction and configuration and a configuration of the front of areconstructing wave by which construction and configurations a lightbeam scanning apparatus can be obtained wherein displacement of ascanning beam position, and a blooming on the scanning surface can beminimized even when the wavelength displacement caused by a wavelengthvariation or wavelength dispersion of the light source arises.

In order to achieve the above objects, a light beam scanning apparatusof the present invention is configured such that a fixed plate, on whicha diffraction grating is recorded, is installed between a rotatablehologram equipped with a diffraction grating and an image formationsurface scanned by this rotatable hologram, wherein:

diffraction gratings are provided in the rotatable hologram and thefixed plate for minimizing a sum total of values obtained by weighting;

a square of the difference between a light flux optical-path length anoptical measured along a principal axis of a light beam incident anddiffracted by the diffraction grating provided in the above-mentionedrotatable hologram, and incident on and diffracted by the diffractiongrating provided in the above-mentioned fixed plate so as to conduct ascanning and converges at a scanning point on an image formationsurface, and a light flux optical-path length measured along a marginalray distanced from the principal axis;

or by weighting an absolute value of this optical path lengthdifference,

the weighting being conducted at every scanning position covering anentire range of an image formation surface.

Further, a light beam scanning apparatus of the present invention isconfigured such that a fixed plate, on which a diffraction grating isrecorded, is installed between a rotatable hologram equipped with adiffraction grating and an image formation surface scanned by thisrotatable hologram, wherein:

diffraction gratings are included in the rotatable hologram and thefixed plate for minimizing a sum total of values obtained by weighting;

a square of a sum is obtained by adding an amount of displacement of alight beam incident on and diffracted by the grating provided in theabove-mentioned rotatable hologram, incident on and diffracted by thegrating provided in the fixed plate so as to perform a scan, andconvergent on a scanning point on an image formation surface, the phasedisplacement of the diffraction grating provided in the rotatablehologram being measured along the peripheral axis distanced from theprincipal axis of an incident reconstructing light flux, to an amount ofdisplacement of the same light. The displacement being measured withrespect to the principal axis of a phase recorded on the diffractiongrating when the light flux is incident on the fixed plate;

or by weighting an absolute value of the above sum,

the weighting being conducted at every scanning position covering anentire range of the image formation surface.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram describing a first embodiment of the presentinvention;

FIG. 2, comprising parts (a)-(c), is a diagram describing the scanningdirection of a light beam scanning apparatus;

FIG. 3 is a diagram describing the cross scanning direction of a lightbeam scanning apparatus;

FIG. 4 is a diagram depicting a configuration of the scanning apparatusof the present invention;

FIG. 5 is a diagram depicting a manufacture of a fixed hologram plate;

FIG. 6, comprising parts (a)-(c), shows graphs describing differencebetween optical path lengths, a beam radius, and a scanning track of thescanning apparatus;

FIG. 7 is a diagram describing waves used for constructing a fixedhologram plate;

FIG. 8, comprising parts (a) and (b), shows diagrams describing spotimages of a scanning beam, which images are obtained by holographicexposure;

FIG. 9, comprising parts (a) and (b), shows graphs describing a scanningtrack and a beam radius when a spherical converging wave is incident;

FIG. 10, comprising parts (a) and (B), shows diagrams describing thescanning direction and the cross scanning direction of the scanningapparatus;

FIG. 11, comprising parts (a)-(c), shows graphs describing a scanningtrack, a beam radius, and displacement due to wavelength variation ofthe scanning apparatus of FIG. 10;

FIG. 12 is a diagram describing a manufacture of the fixed hologramplate;

FIG. 13 is a diagram describing an embodiment of the fixed hologramplate;

FIG. 14 is a diagram describing the conventional technology;

FIG. 15 is a diagram describing the principle of the second embodiment;

FIG. 16 is a diagram showing the first mode of the second embodiment;

FIG. 17 is a diagram showing an example of configuration of a light beamscanning apparatus employing the first mode of the second embodiment (acase where the first hologram is placed to the right of the secondhologram);

FIG. 18 is a diagram showing an example of configuration of a light beamscanning apparatus employing the first mode of the second embodiment (acase where the first hologram is placed so as to be aligned with thesecond hologram);

FIG. 19 is a diagram showing an example of configuration of a light beamscanning apparatus employing the first mode of the second embodiment (acase where the first hologram is placed to the left of the secondhologram);

FIG. 20 is a diagram showing the second mode of the second embodiment;

FIG. 21 is a diagram showing the third mode of the second embodiment;

FIG. 22 is a table showing the relationship between an incident wave andk;

FIG. 23 is a diagram showing an example of an embodiment of a light beamscanning apparatus employing the first through third modes of the secondembodiment;

FIG. 24 is a table showing the relationship among X₁ /F₁, θ₁, θ₂ ;

FIG. 25(A) is a table showing the relationship among W, Δλ, ξ;

FIG. 25(B) shows examples of specifications for designing a secondhologram and a scanning distance thereof;

FIG. 25(C) is a diagram depicting an example of a configuration whereapproximately the same image formation distance is obtained with respectto different outgoing angles;

FIG. 26 is another embodiment of the first mode of the secondembodiment;

FIG. 27 is another embodiment of the first mode of the secondembodiment;

FIG. 28 is another embodiment of the first mode of the secondembodiment;

FIG. 29 is another embodiment of the first mode of the secondembodiment;

FIG. 30 is a diagram depicting an example of a configuration where thelight beam scanning apparatus shown in FIG. 29 is improved;

FIG. 31 is another embodiment of the first mode of the secondembodiment;

FIG. 32 is a diagram depicting an example of a configuration of thelight beam scanning apparatus, which is an improvement on that of FIG.31;

FIG. 33 is a diagram describing a disadvantage of the first embodiment;

FIG. 34 is a diagram describing another disadvantage of the firstembodiment;

FIG. 35, comprising parts (A) and (B), is a diagram describing theprinciple of the present invention;

FIG. 36 is a diagram describing a configuration of the first embodimentof the present invention;

FIG. 37 is a diagram depicting a manufacture of the fixed hologram ofthe first embodiment of the present invention;

FIG. 38, comprising parts (A)-(C), shows a beam intensity distributionof the first embodiment of the present invention (part 1);

FIG. 39, comprising parts (A)-(C), shows a beam intensity distributionof the first embodiment of the present invention (part 2);

FIG. 40 is a diagram depicting a configuration of the second embodimentof the present invention;

FIG. 41, comprising parts (A)-(C), is a diagram for describing acorrection function for ensuring the constant velocity of the fixedhologram plate;

FIG. 42 is a diagram depicting a configuration of the first embodimentof the present invention (cross scanning direction);

FIG. 43 is a diagram depicting a configuration of the first embodimentof the present invention (scanning direction);

FIG. 44, comprising parts (A)-(D), is a diagram describing the fixedhologram plate of the first embodiment of the present invention;

FIG. 45 is a diagram describing the first embodiment of the presentinvention;

FIG. 46, comprising parts (A) and (B), is another diagram describing thefirst embodiment of the present invention;

FIG. 47 is a table showing various beam characteristics obtained whensetting a length of the fixed hologram plate to be short with respect toa light beam scanning distance;

FIG. 48 is a table showing various beam characteristics obtained whensetting a length of the fixed hologram plate to be long with respect toa light beam scanning distance;

FIG. 49 is a diagram depicting a configuration of an embodiment of asixth embodiment;

FIG. 50, comprising parts (A) and (B), shows a configuration and a topview of the first embodiment of the present invention;

FIG. 51, comprising parts (A)-(E), shows graphs for describing anapparatus of FIG. 50;

FIG. 52, comprising parts (A) and (B), is a diagram describing hologramsof the apparatus of FIG. 50;

FIG. 53 is a diagram depicting a configuration of the second embodimentof the present invention;

FIG. 54, comprising parts (A) and (B), shows a side view and top view ofthe second embodiment of the present invention;

FIG. 55, comprising parts (A) and (B), is a diagram describing hologramsof an apparatus of FIG. 53;

FIG. 56 is a diagram depicting a configuration of an optic element,which is an embodiment of the present invention;

FIG. 57 is a diagram describing the principle of the optic element,which is an embodiment of the present invention;

FIG. 58 is a diagram depicting a variation of the optical element shownin FIG. 56;

FIG. 59 is a diagram depicting another variation of the optic elementshown in FIG. 56;

FIG. 60, comprising parts (A) and (B), is a diagram describing theprinciple of a ninth embodiment;

FIG. 61 is a diagram depicting a configuration of an embodiment of thepresent invention (cross scanning direction);

FIG. 62 is a diagram depicting a configuration of an embodiment of thepresent invention (scanning direction);

FIG. 63, comprising parts (A)-(D), is a diagram describing the fixedhologram plate of an embodiment of the present invention (object wave);

FIG. 64 is a diagram describing the fixed hologram plate of anembodiment of the present invention (reference wave);

FIG. 65, comprising parts (A) and (B), shows diagrams describing thefixed hologram plate of an embodiment of the present invention(reference wave);

FIG. 66 are spot diagrams of an embodiment of the present invention;

FIG. 67, comprising parts (A)-(C), shows diagrams describing manufactureof the fixed hologram plate of an embodiment of the present invention(part 1);

FIG. 68, comprising parts (A)-(F), shows diagrams describing manufactureof the fixed hologram plate of an embodiment of the present invention(part 2);

FIG. 69, comprising parts (A) and (B), shows diagrams describing thefirst embodiment of a hologram constructing exposure system of thepresent invention;

FIG. 70 is a diagram describing the second embodiment of the hologramconstructing exposure system of the present invention;

FIG. 71, comprising parts (A) and (B), shows diagrams describing thethird embodiment of the hologram constructing exposure system of thepresent invention;

FIG. 72, comprising parts (A) and (B), shows diagrams depicting a facethologram of the rotatable hologram used in the first embodiment;

FIG. 73, comprising parts (A) and (B), shows diagrams depicting aconfiguration of an embodiment of a tenth embodiment;

FIG. 74, comprising parts (A)-(C), shows diagrams describing amanufacture of a hologram disk of an embodiment of the presentinvention;

FIG. 75 shows spot diagrams of an embodiment of the present invention;

FIG. 76 is a diagram showing a frequency distribution of the rotatablehologram and an incident beam;

FIG. 77 is a diagram describing the principle of an eleventh embodiment;

FIG. 78 is a diagram describing the rotatable hologram of the presentinvention;

FIG. 79 is a diagram describing a configuration of the light beamscanning apparatus of the present invention;

FIG. 80, comprising parts (A) and (B), is a diagram describing an effectof the present invention; and

FIG. 81 is a table showing a changing coefficient determined at thefirst to tenth order.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

A description of the principle of the present invention will be givenbelow, followed by a description of concrete configurations and effectsof the present invention. The first embodiment explained belowconceptually presents a basis for each of the embodiments that will bedescribed herein below.

FIG. 1 is a diagram describing how a correction is carried out for adisplacement of a position of a diffracted light of a scanning opticalsystem, which displacement is caused by a wavelength variation of asemiconductor laser not shown in the figure. A rotatable hologram 10 isequipped with a plurality of diffraction gratings la for carrying out ascanning. An image formation surface 4 is in the form of aphotoconductive drum 3 in such apparatus as a laser printer, forexample. A direction represented by M2, at right angles to a laserscanning direction M3, is called a cross scanning direction. Anassumption here is that a spherical converging wave (hereinafter calledan incident wave 5) having its focus position at MO enters, and that ascanning beam converges, owing to a rotation of the rotatable hologram10, on a scanning point k on the image formation surface 4.Specifically, light flux of the incident wave 5, namely a reconstructinglight, is incident on the rotatable hologram 10 to become a diffractedlight 6, which is further diffracted by a fixed hologram plate 20 tobecome a diffracted light 7, which light 7 converges on the scanningpoint k. An optical path length L_(D) of an optical path originating ineach incident light flux and ending in the scanning point k, the pathlength being measured along a beam, within the incoming wave 5, whoseprincipal axis is a principal axis MA of the rotatable hologram 10, isgiven by the equation (1). Here, the incident light flux is representedas an optical path from the rotatable hologram 10 to the referencesphere whose center is a convergent point P₀. In case of a convergingwave, the sign of an optical path length becomes negative. In theequation shown below, parentheses () indicate a distance between thepoints entered in the parentheses. For example, (A⁰ _(k) P₀) representsa distance between a point A⁰ _(k) and the point P₀. The parentheseshave the same meaning throughout the equations that appear afterequation (1).

    L.sub.0 =-(A.sup.0.sub.k P.sub.0)+(A.sup.0.sub.k B.sup.0.sub.k)+(B.sup.0.sub.k k)                          (1)

An optical path length L₁ of an optical path originating in the incominglight flux of the incoming wave 5, which light flux is incident along amarginal ray MI, and ending in the scanning point k is given as per theequation (2).

    L.sub.1 =-(A.sup.i.sub.k P.sup.i.sub.k P.sub.0.sup.i)+(A.sup.i.sub.k B.sup.i.sub.k)+(B.sup.i.sub.k k)                          (2)

A condition under which the scanning beam is not removed from thescanning point k, even when the incident wave 5 from the semiconductorlaser incurs a wavelength variation at the scanning point k, isrepresented by the equation (3). ##EQU1## That is, the removal isprevented as long as the optical path lengths of the incident wave 5within the light flux, which wave is incoming on courses other thanalong the principal axis MA, are uniform. The condition under which afocal distance on the scanning beam image formation surface 4 does notshow a variation in response to a wavelength variation due to mode hopsof the incident wave 5 from the semiconductor laser 5, is given by theequation (4). ##EQU2## That is, variation is prevented as long as theoptical path lengths of the incident wave 5 remain the same when thewave is incident along the principal axis MA as when it is incidentalong the marginal ray MI. Accordingly, a configuration fulfilling theequations (3) and (4), at the same time, at the scanning point isrequired in order to prevent deterioration caused by a semiconductorlaser wavelength variation in the scanning beam quality across theentire scanning range of the image formation surface 4 on thephotoconductive drum 3.

The above condition of having uniform optical path lengths is met byminimizing and thus optimizing performance functions as per equations(5) or (5-1) below, where the optical path length difference is denotedby δl^(k), and is measured at the scanning point k between the beam thatis incident along the principal axis MA and the beam that is incidentalong the marginal ray MI. ##EQU3## where W_(k) represents a weightdetermined by the degree of minimization of the optical path length ateach scanning point.

A description will be given below of a method utilizing a phase changeby a hologram. In the following, Φ_(in) represents a phase of thewavefront of a wave incoming into the rotatable hologram 10; Φ_(H) ^(k)represents a phase transfer function of the hologram along the principalaxis MA, which hologram is created by the rotatable hologram 10 as therotatable hologram 10 scans the scanning point k; Φ_(H2) ^(k) representsa phase transfer function of the hologram of the principal axiscorresponding to the scanning point k of the fixed hologram plate;δΦ_(H) represents displacement, along the phase of the principal axiswavefront, of the phase of the peripheral wavefront of the incidentlight flux; δΦ_(H) ^(k) represents displacement, from the phase of theprincipal axis wavefront, of the phase transfer function of the hologramcreated by the rotatable hologram 10; δΦ_(H2) ^(k) representsdisplacement, from the phase of the principal axis wavefront, of thephase transfer function of the fixed hologram plate 20.

Since the condition for having regularity in the phases of the incidentbeams on the image formation surface, which regularity is required toform satisfactory images created by an aberration-free scanning beam, inother words to eliminate wavefront aberration, is that the phase of thewave outgoing from the hologram is the sum of the phase of the incomingwave on the hologram and the phase transfer function of the hologram, weobtain the equation (6). ##EQU4## where k₂ represents a wavelength(2π/λ₂)

This equation (6) is transformed into the equation (7) below when theequation (3), which relates to displacement of the scanning beam due toa wavelength variation, and the equation (4), which relates to a focaldistance variation on the image formation surface 4, are both fulfilled.

    δΦ.sup.k .tbd.δΦ.sub.H.sup.k +δΦ.sub.H2.sup.k =0                                                        (7)

This equation (7) is to be fulfilled at the scanning point k forregularity in the phases incident on the image formation surface to beobtained. The equation (7) shows that, in order to maintain a good imageformation quality, the sum of displacements of the phase transferfunctions recorded on the rotatable hologram 10 and the fixed hologramplate 20 should be made zero at each scanning point k. As will be laterdescribed in a detailed description of the equation (7), δΦ^(k) in theequation (7), which represents phase displacement at the scanning pointk, is minimized in a scanning range by using a performance function Eexpressed by the equations (8) or (8-1) below. ##EQU5## where W_(k) is aweight factor introduced in order to reduce displacement of phases ateach scanning point. optimization of a hologram is carried out byminimizing the equations (8) or (8-1).

A description will be given next of a case where the optical pathlengths are uniform. FIG. 2 describes an optical path length along thescanning direction. FIG. 2(a) depicts a parallel wave 6a outgoing fromthe rotatable hologram 10. Given that an incident wave 5a is aconverging spherical wave having a focus MO, the optical path lengths ofthe light beams contained in the light flux, which paths end at eachscanning point on the image formation surface 4, are controlled to beuniform, as can be seen from FIG. 2, by allowing diffracted waves 7a togo out approximately perpendicularly from the fixed hologram plate 20.

FIG. 2(b) depicts a divergent wave 6b, having a focus MO, outgoing fromthe rotatable hologram 10, where an incident wave 5b is a convergingspherical wave as in FIG. 2(a). The optical path lengths of the lightbeams contained in the light flux, which paths end at each scanningpoint on the image formation surface 4, are controlled to be uniform bydirecting the incident wave 7b from the fixed hologram plate 20 to beincident closer to a scanning center than the trajectory of an outgoingwave 6b from the fixed hologram plate 20. The best configuration is theone in which the sign of the diffraction angle is not reversed.

FIG. 2(c) depicts a converging wave 6c outgoing from the rotatablehologram 10, where an incident wave 5c is a converging spherical wave asin FIG. 2(a). The optical path lengths of the light beams contained inthe light flux, which paths end at each scanning point on the imageformation surface 4, are controlled to be uniform by directing anoutgoing wave 7c, outgoing from the fixed hologram plate 20, to beincident closer to the scanning center than the original trajectory of aconverging wave 6c incident on the fixed hologram plate 20, and byreversing the sign of the diffraction angle.

The configurations described above are designed for the scanningdirection; the configurations for the cross scanning direction aredescribed in the following. FIG. 3 is a diagram describing a scanningcarried out in the cross scanning direction, and more particularly aside view showing a configuration by which the optical path lengths inthe cross scanning direction are maintained uniform. A diffracted wave6d is produced from the incident wave 5a incident on the rotatablehologram 10. After being diffracted, the wave outgoing from the fixedhologram plate 20 forms an image on the image formation surface 4 on thephotoconductive drum 3. Parts that are the same as FIG. 3 are given thesame reference notations from figure to figure. In this case, the fixedhologram plate 20 is tilted with respect to the rotatable hologram 10 soas to correct displacement of the scanning beam on the image formationsurface 4 due to a wavelength variation of the reconstructing lightsource, and to obtain the equal optical path lengths. This tilt angle βis configured such that displacement of the scanning beam is minimized.Consequently, when the outgoing wave 6d from the rotatable hologram 10is a parallel wave, the fixed hologram plate 20 allows the wave tofollow, at the scanning end, a trailing trajectory indicated by a brokenline 6'd, so that the optical path lengths of the light flux in eachscanning range are uniform and so that a straight-line scanning on theimage formation surface 4 is possible. The wave is returned to theoriginal image formation point by means of the fixed hologram plate 20,making a straight-line scanning possible. The trajectory of therotatable hologram 10 at the scanning center and the scanning end can beopposite to each other.

As shown in FIG. 2(a), better scanning beam focal-distance correctionfor variation due to a wavelength variation of the reconstructing lightsource, is achieved by making the wavefront of the incident wave 5a,incident on the rotatable hologram 10, be a converging spherical waveand making equal the optical path lengths of the light fluxes, namelythe light flux incident along the principal axis MA and the light fluxincident along the marginal ray MI. The best compensation effect isachieved by making the distance the converging spherical wave 5a travelsbetween the surface of the rotatable hologram 10 and the focal point MOequal, or nearly equal, to the distance between the face of therotatable hologram 10 and the surface of the fixed hologram plate 20.

While the scanning velocity of a normal rotatable hologram 10 becomeshigher as the scanning beam travels to the scanning end when therotatable hologram 10 rotates at a constant angular velocity, thepresent invention allows the scanning beam to be returned to thescanning center through the use of the fixed hologram plate 20, thusmaking it possible to provide both a quantitative matching and acompensation sufficient to make constant the scanning velocity on thescanning surface. Embodiments of a light-beam scanning apparatusemploying a hologram having the above-mentioned attributes are describedin the following.

A description of the first embodiment will be given with reference toFIGS. 4, 5, and 2(a). Referring to FIG. 2(a), in this embodiment, thediffracted wave 6 of the rotatable hologram 10 is a parallel wave 6a,and the outgoing wave outgoing from the fixed hologram plate 20 emergesapproximately perpendicularly from the fixed hologram plate 20.Referring to FIG. 5, a wavelength used in constructing the fixedhologram plate 20 is the same as a wavelength 2 used at the time ofreconstructing. Of the waves used in the construction of the fixedhologram plate 20, an object wave OW is a wave having a sphericalaberration and having a principal axis A of the fixed hologram plate 20as an axial center, which axis is hit, at the scanning center, by theoutgoing wave from the rotatable hologram 10. This object wave is aso-called "positive spherical aberration wave", where a sharper bendtoward the inside is observed away from the axis A and toward the outerboundary. As shown in the equation (11) below, it is best to control thedistance between a point P on the axis, at which point the sphericalaberration wave is supposed to originate, and a point Q on the fixedhologram plate 20, which point Q is hit by the wave, namely the opticaldistance (PQ), to be of a predetermined distance (d) at any point.

    d=P.sub.0 Q.sub.0 =P.sub.1 Q.sub.1 =P.sub.2 Q.sub.2 = . . . =P.sub.n Q.sub.n(9)

A reference wave RW is a parallel wave incoming diagonally and having anincidence angle α (≠0). The above-mentioned parameters d and α aredetermined as appropriate so that the aforementioned performancefunctions (5) or (8) are fulfilled, aberration is reduced, and a linearscanning can be performed.

The values in FIG. 6 are obtained by realizing the above settings. FIG.6(a) represents a result of optimization using the equation (5), wherethe horizontal axis indicates a scanning width occurring when thescanning center of the photosensitive drum is designated as 0.0, and thevertical axis indicates an optical path length difference. This graphtells that the optical path length difference between the outermostbeams of the light flux, which difference is measured in the scanningdirection when the scanning width is 108 mm (A4 size scan), has themaximum value of 30 λ. This value translates into a distance of 0.03 mm.Since the total optical path in this case is 641 mm, these constitutepractically regular optical paths, that is, no optical path lengthdifference, results. In this case, the wavelength of the reconstructingwave generated by the semiconductor laser is λ2=780 nm, the rotatablehologram 10 has a regular pitch, and the spatial frequency thereof is1765 (pcs/mm). The angle of the beam incident on the rotatable hologram10 is 44.2°, and the radius of the rotatable hologram 10 is 40 mm. Asfor the parameters of the fixed hologram plate 20, d=364 mm, and α=6.5°.The distance between the rotatable hologram 10 and the fixed hologramplate 20 is 218 mm, and the distance between the fixed hologram plate 20and the image formation surface 4 is 360 mm.

The tilt angle of the fixed hologram plate 20 with respect to therotatable hologram 10 was 45.0° in order to fulfill the performancefunction (8). FIG. 6(b) shows the scanning beam characteristic obtainedtherefrom. That is, for the scanning width of 216 mm, the beam radius iswithin 18 μm. As shown in FIG. 6(c), a deviation from a straight line ofbelow the ±78 μm level and a linearity of below the ±0.12% levelresulted. Moreover, the variation of wavelength of the semiconductorlaser was controlled to be less than 1 μm in the scanning direction evenin the presence of a 0.3 nm wavelength variation due to a mode hop. Asshown in FIG. 3, the scanning beam from the rotatable hologram 10 andincoming into the fixed hologram plate 20 was bent in a simple manner soas to obtain a straight-line scanning on the fixed hologram plate 20,with the result that a displacement of 1 mm was observed.

Once the interference pattern on the fixed hologram plate of thisembodiment is determined, the pattern can be drawn with an electron beamor a laser plotter. This method of manufacturing a fixed hologram plateby holographic exposure will be described in the following.

It is generally known that wavelength sensitivity of a hologram materialhaving a high diffraction efficiency is in a range shorter than that ofthe wavelength of a semiconductor laser. Thus, aberration owing to thiswavelength ratio must generally be taken into consideration whenmanufacturing a hologram plate by holographic exposure. Here, thewavelength of the wave used in constructing the hologram plate isdesignated as λ1 and the wavelength ratio is designated as λ2/λ1. It isfound that, after taking into consideration aberration owing to thiswavelength ratio, the construction of a spherical aberration needed forthe construction of a hologram wave can be such that d of the firstembodiment is replaced by the product of d and s. As in the firstembodiment, optimization was carried out by employing a diagonallyincident, parallel reference wave. Once the relevant interference fringedistribution is known, a hologram, containing aberration of the abovecomplexity, needs to be manufactured by holographic exposure.

FIG. 7 shows a second embodiment of the fixed hologram plate 20 of thepresent invention, wherein a spherical aberration wave used therein isof a wavefront of a wave outgoing from a plano-concave lens, which is aspherical lens, on which plano-concave lens a stigmatic divergingspherical wave is incident. Parameters, including a plano-concave lens,are optimized so that the above amount of aberration is obtained. Thatis, λ1=441.6 nm (HeCd laser), and the wavelength of the semiconductorlaser is designed to be λ2=780 nm. The thickness of the BK7plano-concave lens is 3.0 mm at the center, an index of refraction is1.51, and a curvature thereof is 115.0 mm. The distance between thepoint light source S₀ of the diverging spherical wave and theplano-concave lens is d₀ =439.0 mm, and the distance between theplano-concave lens and the fixed hologram plate 20 is LT1=469.0 mm.

FIG. 8(a) shows aberration images of the scanning beam created by thefixed hologram 20 manufactured by holographic exposure designed inaccordance with FIG. 7. In the figure, a very small aberration of belowa 20 μm level is evident. FIG. 8(b) also shows spot aberration images ofthe scanning beam created by the hologram manufactured in accordancewith FIG. 5. In FIG. 8(b), approximately the same images as in FIG. B(a)are seen. This embodiment has an advantage in that holographic exposurecan be achieved by a simple spherical aberration wave and control of theexposure system is fairly easy.

In a third embodiment described below, the incident wave which isincident on the rotatable hologram 10 is a converging spherical wave.The phase transfer function in this case needs to fulfill the followingequation (10). ##EQU6## It is evident here that a point light source ofthe reference wave is at the distance F₁ from the rotatable hologram.The distance F₁ is measured along an axis of rotation of the rotatablehologram and the wavelength of the wave used in constructing a hologramis λ_(1'). λ_(1') here is a virtual wavelength of the wave used inconstructing a hologram. The object wave is a spherical wave produced bya point light source positioned at a distance Y₂ and a height F₂ /S fromthe rotatable hologram, which distance is measured along an axis ofrotation aligned with the principal scanning axis. The wavelength of thewave used in constructing a hologram is a virtual wavelength λ_(1").Thus the virtual difference is provided, in terms of the wavelength,between the reference wave and the object wave, which are both used inconstruction of a hologram. S is the ratio λ₂ /λ_(1") obtained from λ₂and λ_(1") of the reconstructing wave. Optimization in accordance withthe equation (5) was conducted in a scanning apparatus equipped with therotatable hologram 10 manufactured on the basis of the equation (10), onan assumption that a converging spherical wave is incident on therotatable hologram.

The result of this arrangement is that a deviation from a straight lineof below the ±0.1 mm level was obtained, as shown in FIG. 9(a). FIG.9(b) shows that a beam radius of less than 18 μm was obtained. Alinearity of below ±0.22% level resulted. As for displacement in thescanning direction due to a wavelength variation of the semiconductorlaser, a displacement of less than 1 μm in correspondence with a 0.3 nmwavelength variation was observed, which is a satisfactory result. Inthis arrangement, the radius of the beam incident on the rotatablehologram 10 is 45 mm, the distance between the rotatable hologram 10 andthe fixed hologram plate 20 is 182 mm, the distance between the fixedhologram plate 20 and the image formation surface is 277 mm, and thetilt angle of the fixed hologram plate 20 with respect to the rotatablehologram 10 is 64.2°. As far as the reference wave is concerned, λ_(1')=330 nm and F1=200 mm. With the object wave, λ_(1") =78 nm, andtherefore S=10, F2=1060, and Y2=95 mm.

This embodiment is configured such that the incident wave is aconverging-spherical wave and that the distance between the surface ofthe rotatable hologram and the convergent point is 200 mm, which isapproximately the distance between the surface of the fixed hologramplate and the image formation surface. Even when a variation of 100 nmis caused for environmental reasons in the wavelength of thesemiconductor laser, the beam radius incurred only a minor change from18 μm to 18.5 μm, meaning that no serious deterioration in the beamradius takes place. While the configuration of a hologram represented bythe equation (10) assumes that the wavelength of the wave used in themanufacture of a hologram is a virtual wavelength, the manufacture of ahologram by an electron beam or a laser plotter drawing is possible.When the manufacture is conducted using holographic exposure, anauxiliary optical system proposed in the Japanese Laid-Open PatentApplication 63-72633 filed by the present applicant can be utilized.

A fourth embodiment will be described below. FIGS. 10(a) and (B)illustrate a compensation for displacement of the scanning beam, whichdisplacement occurs when a removal of the base of the rotatable hologram10 from a parallel state occurs. In this fourth embodiment, a beam,whose convergence takes place in the cross scanning direction at rightangles to the scanning direction (a direction of rotation of therotatable hologram 10), is employed as the beam incident on therotatable hologram 10, as shown in FIGS. 10(a) and (B). Since the waveincident on the fixed hologram plate 20 is a cylindrical wave, areference wave that matches this cylindrical wave is considered to benecessary. This means that a spherical aberration wave as the one inFIG. 5 is to be used as the object wave for the manufacture of the fixedhologram plate 20. For the reference wave, one example is a wave havingdirection cosines as per the equation (11) below. ##EQU7## where C₀, y₀,and Z₀ are constants.

While the reference wave is a coma wave as shown by the equation (11)above, the object wave is a spherical aberration wave. This aberrationcan be controlled to be at an appropriate level so that a desiredperformance can be obtained. FIG. 11 shows the result thereof. FIG.11(a) illustrates the deviation from a straight line, while FIG. 11(b)illustrates the beam radius.

The present invention realizes an extremely satisfactory deviation froma straight line of below the ±0.4 m level. The beam radius thereof is 8μm at a maximum, which is sufficient to allow a successful aberrationcorrection. Linearity is below ±0.13% level, which is also satisfactory.As shown in FIG. 11(c), even under a wavelength variation of thesemiconductor laser of 1 nm, for example, the displacement could becontrolled to be less than 3 μm in the scanning direction, and less than3 μm in the cross scanning direction. The relationships among the objectwave, the reconstructing wave, the parameters of the rotatable hologram10, and the fixed hologram plate are of the same parameters as those inthe first embodiment. Also, y₀ =-5 mm, and Z₀ =321 mm.

The base of the rotatable hologram in this embodiment can be moved fromits ideal position and still function well in the following way. Thatis, even the rotatable hologram 10 exhibiting a displacement as large asone minute (P--P) from its ideal position allows a sufficient correctionin which the displacement in the cross scanning direction is controlledto be less than 5 μm. This means a greater tolerance compared to theconventional rotatable hologram 10, where only several seconds ofdisplacement was allowed from the ideal position of the base, and goes along way toward reducing the cost of a hologram base.

As shown in FIG. 12, when manufacturing a fixed hologram plate, aspherical aberration wave for the object wave is generated by aspherical lens, and the reference wave is generated by a similarspherical lens 8 that creates direction cosines of a coma wave 9 asrepresented by the equation (11).

FIG. 13 depicts the fifth embodiment. Two holograms are formed on onefixed hologram plate. The above-described object wave for manufacturingthe fixed hologram, and the wavefront C are recorded on the fixedhologram plate so that one hologram 20-1 is manufactured. The otherhologram 20-2 is manufactured with the above-described reference wavefor manufacturing the fixed hologram plate and with the wavefront C. Bysuperposing, as shown in the figure, characteristics similar to those ofthe above embodiments are obtained. This embodiment is most suitable forthe case where the fixed hologram plate is almost of an in-line type andholographic exposure is difficult.

Since each of the holograms thus manufactured is of an off-axis type, ahigh diffraction efficiency results. Further, these two hologram platesachieve regular optical paths and precise compensation for degradationof characteristics of the scanning light; which degradation is due tovariation of the wavelength of the semiconductor laser. The fixedhologram plates here are mass producible by means of injection, makingthis embodiment favorable in terms of manufacturing and pricing. Theshape of the rotatable hologram is not limited to a disk, and thepresent invention is applicable to other shapes including a cylinder, acone, and a pyramid.

As has been described, the first invention is capable of providing asimple and inexpensive optical system with two holograms. Ahigh-reliability optical system without displacement of the scanningbeam, which displacement is due to variation of the wavelength of thesemiconductor laser, is realized in the above invention.

The second invention included in the present application will bedescribed now with reference to FIGS. 15 and 16. FIG. 15 is an obliqueview illustrating a light beam scanning apparatus according to the firstmode of the present invention. FIG. 16 is a top view thereof. The lightbeam scanning apparatus 100 comprises at least two holograms, namely afirst hologram plate 110 and a second hologram plate 112. 120 representsa scanning surface.

The first hologram plate 100 is a movable hologram for converting aconverging spherical wave into a parallel wave, for example. The secondhologram plate 112 is a fixed hologram for converting a parallel waveinto a converging spherical wave, for example. The distance between thefirst hologram plate 110 and the convergent point is denoted by F₁, thedistance between the second hologram plate 112 and the convergent pointis denoted by F₂, and the distance between the first hologram plate 110and the second hologram plate 112 is denoted by L. A central wavelengthof the light source is indicated by λ.

It is ensured in this configuration that an incident beam having a beamradius of W is perpendicularly incident on the first hologram plate 110and is diffracted at an angle θ₁, after which the beam is incident onthe second hologram plate 112 disposed to be parallel to the firsthologram plate 110, wherefrom it is diffracted at an angle θ₂, allowingan image to be formed on the scanning surface 120 disposed at thedistance F₂ from the second hologram plate 112.

Designating spatial frequencies of the first hologram 110 and the secondhologram 112 as f₁ and f₂ respectively, the following equation isderived.

    sin θ.sub.1 =f.sub.1 λ

    sin θ.sub.1 + sin θ.sub.2 =f.sub.2 λ

Therefore,

    sin θ.sub.2 =(f.sub.2 -f.sub.1)λ

Providing that a displacement of Δλ is created in the central wavelengthof the light source, the following equations are derived.

    cos θ.sub.1 ·Δθ.sub.1 =f.sub.1 Δλ

    cos θ.sub.1 ·Δθ.sub.1 + cos θ.sub.2 ·Δθ.sub.2 =f.sub.2 Δλ

Therefore,

    cos θ.sub.2 ·Δθ.sub.2 =(f.sub.2 -f.sub.1)Δλ

    = sin θ.sub.2 (Δλ/λ)

A displacement ΔX of the scanning beam, created by a wavelengthvariation, is as follows.

    ΔX=Δθ.sub.2 (F.sub.2 / cos θ.sub.2)/ cos θ.sub.2

    =F.sub.2 sin θ.sub.2 (Δλ/λ)/ cos.sup.3 θ.sub.2                                             (21)

The scanning beam diameter D is determined by an F number and anaperture W as follows. ##EQU8## where k is a constant.

A displacement ΔX of the scanning beam position is required to be lessthan 1/4 of the scanning beam diameter in a light-scanning apparatussuch as used in a printer, in order to maintain a satisfactoryresolution. Accordingly, ΔX/D is obtained from the equations (21) and(22) as follows.

    ΔX/D= sin θ.sub.2 ·(W/k)(Δλ/λ.sup.2)<1/4       (23)

An optical path length difference ΔΦ₁ is obtained as W· sin θ₂.Therefore, it is derived using the equation (23) that

    ΔΦ.sub.1 =W sin θ.sub.2 <(1/4)k(λ.sup.2 /Δλ)                                         (24)

Since generally k˜2,

    ΔΦ.sub.1 <C(λ.sup.2 /Δλ)     (25)

where C is a constant smaller than 0.5.

As is evident from the above, it is required that the optical pathlength of the scanning beam be smaller than C (λ² /Δλ) in order toobtain sufficient resolution with a light scanning apparatus undervariations of the wavelength.

A similar condition is derived with regard to a blooming of the beam,which blooming is caused by the variation of the wavelength. Accordingto Rayleigh's resolution, a wavefront aberration small enough not tocause a blooming is λ/4. When a variation of the wavelength is Δλ andthe optical path length difference of the scanning beam is denoted byΔΦ₂, the wavefront aberration is expressed by

    ΔΦ.sub.2 (Δλ/λ)<(λ/4)

Therefore,

    ΔΦ.sub.2 <(1/4)(λ.sup.2 /Δλ) (26)

It results from the equations (25) and (26) above that the optical pathlength difference ΔΦ of the scanning beam should fulfill therelationship shown below in order to maintain a regular resolution undera wavelength variation or wavelength dispersion.

    ΔΦ=ΔΦ.sub.1 +ΔΦ.sub.2 <C(λ.sup.2 /Δλ)                                         (27)

where C is a constant. Accordingly, a light beam scanning apparatusfulfilling the equation (27) does not allow a displacement of thescanning beam, a blooming, or a displacement of a focus even in thepresence of a wavelength variation of the light source.

FIGS. 17 through 19 illustrate examples of configurations of a lightscanning apparatus fulfilling the equation (27). Referring to FIGS. 17through 19, a movable first hologram plate 110 and a fixed secondhologram plate 112 are each separated into seven segments.

Referring to FIG. 18, when the first hologram plate 110 is positioned atx₄, the incident beam is diffracted by the segment 4 of the firsthologram plate 110, and is diffracted by the segment 4' of the secondhologram plate 112 before reaching a point P₄. These two hologram plates110 and 112 are disposed such that the optical path length difference ofthe beams is smaller than (1/2) (λ² /Δλ).

When the first hologram plate 10 is moved to the right, as shown in FIG.17, the incident beam is diffracted by the segment 1, and diffracted bythe segment 1' of the second hologram plate 112 before reaching a pointP₁.

Similarly, referring to FIG. 19, the incident light is diffracted by thesegment 7 and the segment 7' before reaching a point P₇.

The first hologram plate 110 and the second hologram plate 112 aredisposed such that, in every case of diffraction taking place at an Mthsegment and at an M'th segment, the optical path length difference ofthe beam is smaller than (1/2) (λ² /Δλ).

By moving the hologram plate 110 back and forth in the abovelight-scanning apparatus, a digital light scanning apparatus can beobtained which is free from a displacement of the scanning beamposition, a blooming of the scanning beam, and displacement of a focus,even under a wavelength variation Δλ.

Next, the principle of a light-scanning apparatus according to thesecond mode of the second invention will be described with reference toa top view of a typical hologram scanning system as shown in FIG. 20. InFIG. 20, configurations that correspond to those in FIGS. 15 and 16 aregiven the same reference notations. The light beam scanning apparatus100 of FIG. 20 consists of at least two hologram plates, namely a firsthologram plate 110 and a second hologram plate 112.

A converging spherical wave is incident, with an incidence angle α, onthe first hologram plate 110 (having a focal distance l₁) of the lightbeam scanning apparatus 100 shown in FIG. 20. This converging sphericalwave is converted into a diverging spherical wave by the first hologramplate 110 and goes out from the first hologram plate 110 at an outgoingangle of δ. The second hologram plate 112 is disposed to be parallel tothe first hologram plate 110 and is separated therefrom by an opticalaxis distance 1₄ (the distance between H₁ and H₂ in FIG. 20).

The diverging spherical wave having a focal distance l₃ (the distancebetween H₂ and H₃ in the figure) and outgoing from the first hologramplate 110 is incident, at an incidence angle δ, on the second hologramplate 112, where the wave is converted into a converging spherical wave.This converging spherical wave, having a focal distance l₂ (the distancebetween H₃ and H₄ in the figure), exits the second hologram plate 112 atan outgoing angle β.

In order to control the optical path length difference to be 0, in otherwords to obtain an achromatic condition with regard to an optical axisand an image formation, the following equations must hold.

    sin α={l.sub.3 /(l.sub.3 =l.sub.4)} sin β-{l.sub.4 /(l.sub.3 -l.sub.4)} sin δ                                    (28)

    cos.sup.2 α/2/l.sub.1 ={l.sub.4 /l.sub.3 -1.sub.4) .sup.2 /2} cos.sup.2 δ+{l.sub.3.sup.2 /(l.sub.3 /l.sub.4).sup.2 /l.sub.2 /2} cos.sup.2 δ                                         (29)

Assuming that the optical axis of the spherical wave incident on thefirst hologram plate 110 is perpendicular, that is α=0, and calling

    k(δ)=l.sub.4 /l.sub.3                                (30)

the equation (28) becomes

    sin β=k(δ) sin δ                          (31)

k (δ) can be regarded as a parameter representing a degree to which thescanning beam outgoing from the first hologram plate 110 is dispersed.

Accordingly, in a light beam scanning apparatus fulfilling the equation(31), the optical path length difference ΔΦ of the scanning beam can becontrolled to be 0, which difference is required in order to fulfill theequation (27) explained in the description of the first mode of thisinvention.

Therefore, in accordance with the second mode of the present invention,a light scanning apparatus of even better performance than the lightscanning apparatus of the first mode of this invention can be obtained,wherein the light scanning apparatus is free from any displacement ofthe scanning beam position, blooming of the scanning beam, ordisplacement of a focus even under a wavelength variation of thereconstructing-beam light source.

If we approximate k(δ) by expanding it with respect to δ,

    k(δ)=k.sub.0 +k.sub.1 δ.sup.2                  (32)

Since the spatial frequency f(x) of the second hologram plate 112 is(sin δ-sin β)/λ, the following equation holds.

    λf(x)={1-k(δ)} sin δ

    =(1-k.sub.0 -k.sub.1 δ.sup.2) sin δ            (33)

Since the first-order differential f' (x) of the spatial frequency f(x)is tan δ=x/L when the first hologram plate 110 and the second hologramplate 112 are separated by the distance L (L=1₄), it is found from theequation (31) that

    λf'(x)={-2k.sub.1 δ sin δ+(1-k.sub.0 -k.sub.1 δ.sup.2) cos δ}× cos.sup.2 δ/L    (34)

Designating, in the second hologram plate 112, the distance between theobject point of the incident wave and the second hologram plate 112, andthe distance between the image point and the second hologram plate 112as, respectively, a(δ) and b(δ), the following relationship regardingthe image formation is derived.

    cos.sup.3 /a(δ)+ cos.sup.3 β/b(δ)=λf'(x)(35)

At the scanning center (δ→0), it is found from the equations (34) and(35), that

    1/a(0)+1/b(0)=(1-k.sub.0)/L                                (36)

Referring to the above equation, a plane image formation is obtainedwhen b(δ)=b(0)=b₀. Further, the following relationship is obtained.

    k(δ)=l.sub.4 /l.sub.3 =L/a(δ)                  (37)

Expanding a(δ) in the power series of ##EQU9##

It is found from the equation (37) that

    1/a(δ)=k(δ)/L

    =(k.sub.0 +k.sub.1 δ.sup.2)/L                        (39)

Comparing the equations (38) and (39), we obtain:

    1/b.sub.0 =(1-2k.sub.0)/L                                  (40)

    k.sub.1 =-(3/8) (1-k.sub.0.sup.2)(1-2k.sub.0).             (41)

By determining k(δ) and a(δ) according to the equations

    k(δ)=k.sub.0 -(3/8) (1-k.sub.0.sup.2)(1-2k.sub.0)δ.sup.2(42)

    a(δ)=L/k(δ)                                    (43)

a light scanning apparatus having no optical axis displacement due to awavelength variation of the light source may be obtained, where k₀ is aparameter specifying the characteristic of an optical system. Since itis assumed in the equation (40) that b₀ is positive and k(δ) is morethan 0,

    0<k.sub.0 <0.5.                                            (44)

The converging spherical wave incident on the first hologram plate 110is defined by the equation (29) by assuming α=0 and by using theequations (30) and (32). l₁ concerns the scanning beam incident on thehologram plate 110 and is generally a constant. Therefore, it issufficient to consider the case of δ=0 only. As a result of theseconsiderations, l₁ is determined from

    1/l.sub.1 =(k.sub.0.sup.2 /L+1/b.sub.0)/(1-k.sub.0).sup.2.

It is known from the equation (40) that

    l.sub.1 =L                                                 (45)

From these results, the following conditions for an achromatic lightscanning apparatus are derived.

1 It is required that the wave incident on the first hologram plate 110be a converging spherical wave in order to fulfill conditions forachromatic image formation.

2 It is required that either the spatial frequency f(x) of the secondhologram plate 112 be

    f(x)={1-k(δ)} sin δ,

where δ is the incidence angle of the beam incident on the secondhologram plate 112, or that

    k(δ)=k.sub.0 -(3/8)(1-k.sub.0.sup.2)(1-2k.sub.0)δ.sup.2 ;

3 k(δ) of the above equation serves to determine the position of thediverging incident-wave light source with respect to the second hologramplate 112. In order to fulfill the conditions for achromaticity relativeto the optical axis, it is required that

    a(δ)=L/k(δ).

4 The scanning beam outgoing from the second hologram plate 112 forms aplane image in accordance with the following equation.

    1/b.sub.0 =(1-2k.sub.0)/L

A description will be given next of the principle of a light beamscanning apparatus according to the third mode of the second inventionwith reference to a top view of a typical hologram scanning system asshown in FIG. 21. The light beam scanning apparatus 100 of FIG. 21consists of at least two holograms, namely a first hologram plate 110and a second hologram plate 112.

A scanning beam in the form of a converging spherical wave is incidenton the first hologram plate 110 (having a focal distance of l₁) of thelight beam scanning apparatus shown in FIG. 21 at an incidence angle ofα. This converging spherical wave is converted into a convergingspherical wave by the first hologram plate 110 and exits therefrom at anoutgoing angle of δ.

The second hologram plate 112 is disposed to be parallel to the firsthologram plate 110 and at an optical axis distance of l₄ (the distancebetween H₁ and H₃ in FIG. 21) therefrom. The converging spherical waveoutgoing from the first hologram plate 110 having a focal distance (l₃-l₄) (l₃ minus l₄) (the distance between H₂ and H₃ in the figure) isincident, on the second hologram plate 112, with an incidence angle δ.It is then converted into a converging spherical wave by the secondhologram plate 112. This converging spherical wave having a focaldistance of l₂ (the distance between H₃ and H₄ in the figure), exitsfrom the second hologram plate 112 with an outgoing angle of β. Adistance between H₁ and H₂ in FIG. 7 is l₃.

The following equations must stand in order for the optical path lengthdifference of the scanning beams to be 0, in other words in order for anachromaticity to be obtained with respect to an optical axis and to animage formation.

    sin α=(1-l.sub.4 /l.sub.3) sin β+(l.sub.4 /l.sub.3) sin δ(46)

    cos.sup.2 α/2/l.sub.1 ={l.sub.4 /l.sub.3.sup.2 /2} cos.sup.2 δ+{(l.sub.3 -l.sub.4).sup.2 /l.sub.2 /l.sub.3.sup.2 /2} cos.sup.2 β                                                    (47)

By calling k' (δ)=l₄ /l₃, the following conditions for obtaining anachromatic light beam scanning apparatus can be derived, as in the caseof the second mode of the light beam scanning apparatus of the presentinvention discussed earlier.

1 It is required that the incident wave incident on the first hologramplate 110 be a converging spherical wave in order to fulfill theconditions for achromatic image formation.

2 It is required that either the spatial frequency f(x) of the secondhologram plate 112 be

    f(x)={1-k(δ)} sin δ,

where δ is the incidence angle of the beam incident on the secondhologram plate 112, or that ##EQU10##

3 where k' (δ) of the above equation serves to determine the position ofthe diverging incident-wave light source with respect to the secondhologram plate 112. In order to fulfill the conditions for achromaticityrelative to the optical axis, it is required that ##EQU11## The negativevalue of a₀ indicates that the beam incident on the second hologramplate 112 is a converging spherical wave.

4 The scanning beam outgoing from the second hologram plate 112 forms aplane image in accordance with the following equation.

    1/b.sub.0 =(1+k'.sub.0)/(1-k'.sub.0)/L

5 Since b₀ in the above equation is positive, it is required that 0<k'₀<1.

As is evident from the above, the requirement for k' relevant to thethird mode of this invention, where the light outgoing from the secondhologram plate 112 is a converging spherical wave, is

    k'(δ)=k(δ)/{k(δ)-1}

where k can be expanded from the range 0-0.5 to the range -∞-0.5.Generalization is possible by calling η=1-k, η'=1-k', η=1/η'.

FIG. 22 shows modes of the incident and outgoing waves of the firsthologram plate and modes of the outgoing waves of the second hologramplate with respect to variation of the factor k in the configurationsshown in FIGS. 17-20.

FIG. 23 shows the first embodiment of a translation light beam scanningapparatus according to the above-described first mode. A movable firsthologram 150 is linearly driven in the direction X in the figure bymeans of a translation mechanism 160 such as a voice coil motor. Thedirection on a hologram surface perpendicular to the direction X isdesignated as the direction Y.

The first hologram 150 and the second hologram 112 disposed and fixed ata distance L therefrom are configured such that the spatial frequencydistributions f_(x) (the direction X) and f_(y) (the direction Y) aregiven by the following equations.

The first hologram 50:

    f.sub.x λ=x.sub.1 /F.sub.1

    f.sub.y λ=y.sub.1 /F.sub.1                          (48)

The second hologram 12:

    f.sub.x λ=x.sub.2 /F.sub.2

    f.sub.y λ=y.sub.2 /F.sub.2                          (49)

The suffixes here represent coordinates on the first hologram plate 150and on the second hologram plate 112.

When a converging spherical wave having a focal distance of F₁ isincident perpendicularly on the movable first hologram plate 150, thescanning beam in the form of roughly a plane wave outgoes from the firsthologram plate 150. This beam is diffracted by the fixed second hologramplate 112, causing the beam to converge on a scanning surface 120. Whenthe first hologram plate 150 is moved a distance x₁, the beam outgoingfrom the first hologram plate 150 is subject to an angle change by θ₁=sin⁻¹ (x₁ /F₁). The outgoing beam is incident on the second hologramplate 112 in such a way as to form an angle θ₁ with respect to thelength L tan θ₁, the beam then outgoes from the second hologram plate atan angle θ₂, wherein ##EQU12##

It is found from the equation (27) that the optimum condition for awavelength variation Δλ is given by

    ΔΦ=W· sin θ.sub.2 <(1/4)(λ.sup.2 /Δλ)

Therefore,

    sin θ.sub.2 =L tan { sin.sup.-1 (x.sub.1 /F.sub.1)}/F.sub.2 -x.sub.1 /F.sub.1 <(1/4W)(λ.sup.2 /Δλ).tbd.ξ(50)

    x.sub.2 =L tan θ.sub.1                               (51)

Calling L=F₂, the relationship among x₁ /F₁, θ₁, and θ₂ is tabulated inFIG. 24. ξ=(1/4W) (λ₂ /Δλ) is tabulated in FIGS. 25(A)-25(C), assumingλ=780 nm.

A scanning optical system resistant to wavelength variation can beconfigured on the basis of the values of sin θ₂ in FIG. 24 and those ofξ in FIG. 25(A).

Take a light beam scanning apparatus, for example, where a semiconductorlaser having a wavelength variation of 1 nm is used to constructholograms characterized by the equations (48) and (49) in which F₁ =F₂=200 mm, where the distance L between the first hologram plate 150 andthe second hologram plate 112 is 200 mm, and the beam radius of thereconstructing beam is 2 mm (F number being 100). ξ in this case is0.076 according to FIG. 25(A).

When the first hologram plate 150 in this configuration is translated±100 mm (x₁ /F₁ =0.5), it is found from FIG. 24 that sin θ₂ =0.77 andthe equation (50) roughly stands valid. As a result, a light scanningapparatus having a scanning width (twice x₂) of 252 mm (B4 size) isobtained, where no degradation of the scanning performance due todisplacement of the beam position caused by a wavelength variation isobserved.

The second embodiment based on the first mode of this invention isillustrated in FIG. 26. In this embodiment, a rotary hologram disk 150is used as the first hologram. As in the first embodiment, this firsthologram is combined with a fixed second hologram plate 112 so that theconditions specified by the equation (27) are met with a configurationof the movable first hologram plate 150 and the fixed second hologramplate 112.

A preferred embodiment is achieved by disposing the second hologramplate 112 to be parallel to the first hologram plate 150, and byensuring that the principal axis 0 of the hologram of the secondhologram plate 112 passes a point, at which point the beam is outgoing,of the first hologram plate 150.

A third embodiment according to the first mode of this invention isillustrated in FIG. 27. A rotary truncated-cone hologram 150 is used asthe movable first hologram. A hologram is created on the surface of thetruncated cone by means of a spherical wave and a plane wave.Preferably, a wavefront having its center on the rotation axis of thecone is created by a plane wave perpendicular to the surface of thetruncated cone.

The fixed second hologram plate 112 is tilted at roughly the same angleas the truncated cone surface of the first hologram plate 150.Preferably, they are disposed so as to be parallel to each other. Byarranging the first and second holograms 150 and 112 in such a way thatthe equation (27) is fulfilled, a light beam scanning apparatusresistant to wavelength variations is obtained.

A fourth embodiment according to the first mode of this invention, inwhich mode a cylindrical hologram 150 is used as the movable firsthologram, is illustrated in FIG. 28. A hologram is created on thesurface of the cylinder by a spherical wave and a plane wave.Preferably, a wavefront having its center on the rotation axis of thecylinder is created by a plane wave.

The fixed second hologram plate 112 is preferably disposed so as to beparallel to the cylinder surface. By arranging the first and secondhologram plates 112 and 150 in such a way that the equation (27) isfulfilled, a light beam scanning apparatus resistant to wavelengthvariations is obtained.

FIG. 29 illustrates a fifth embodiment of an electronic light beamscanning apparatus according to the first mode of this invention. Amovable first hologram 110 is formed of an acoustic optic element.Typically, such an element is manufactured by creating a diffractiongrating comprising interference fringes having a pitch d of 9-18 μm byapplying a high-frequency electrical field having a center frequency ofabout 55 MHz to a tellurium oxide crystal, and subjecting the crystal to±18 MHz frequency modulation. The light scanning beam is diffracted asthese fringes change with the passing of time.

Designating the direction perpendicular to the direction X (X is thedirection of the movement of the first hologram) as the direction Y, thefirst hologram 110 and the second hologram 112 disposed and fixed at adistance L therefrom are configured such that the spatial frequencydistributions f_(x) and f_(y) are given by the following equations.

    f.sub.x λ=A sin ωt+B, f.sub.y λ=0

(The first hologram 110)

    f.sub.x λ=x.sub.2 /F.sub.2,f.sub.y λ=Y.sub.2 /F.sub.2

(The second hologram 112)

When a plane wave beam is perpendicularly incident on the first hologram110, a scanning beam in the form of a plane wave exits therefrom. Thisbeam is diffracted by the second hologram plate 112, and converges onthe scanning surface 120. The outgoing beam diffracted by the firsthologram 110 is deflected by an angle θ₁ = sin⁻¹ (A sin ωt+B). Thisoutgoing beam is incident, at an incidence angle of θ₁, on a position Ltan θ₁ on the second hologram plate 112, and exits from the secondhologram plate 112 at an angle θ₂.

It results from this configuration that ##EQU13##

It is found from the equation (27) that the condition needed to enable adevice to be protected against wavelength variations is as follows.

    ΔΦ=W sin θ.sub.2 <(1/4)(λ.sup.2 /Δλ)

Therefore, the following relationship holds.

    |L tan { sin.sup.-1 (A sin ωt+B) }/F.sub.2 -(A sin ωt+B)|<(1/4W)(λ.sup.2 /Δλ)

When sin θ₁ is relatively small, the left side of the above inequalitybecomes |(L/F₂ -1) sin θ_(1max) |·θ_(1max) is about 5.

Assuming that a semiconductor laser having a center wavelength of 780nm, and a wavelength displacement of 5 nm is employed as the lightsource 114, and that the incident beam radius W is 5 mm, then we obtainthe following.

    (1/2W)(λ.sup.2 /Δλ)=0.012

Accordingly, it is required that L/F₂ =1±0.14. When an actual attemptwas made by the inventor to configure a light beam scanning apparatus inwhich L=100 mm and F₂ =110 mm, a light beam scanning apparatus resistantto wavelength variation was obtained.

Although FIG. 29 illustrates an embodiment in which a single firsthologram is used, it is also possible to alternatively dispose aplurality of holograms such as the first hologram in stages.

FIG. 30 illustrates a sixth embodiment of an even more efficientconfiguration wherein an incident beam is provided in the form of aparallel light so as to prevent a blooming of the scanning beam, theparallel light being created by disposing a converging lens 116 and amagnifying hologram lens 118 in the stage preceding the aforementionedfifth embodiment and subsequent to the diverging light source.

In the fifth embodiment, an optical path length difference ΔΦ is createdbetween the center of the scanning beam and the end thereof so as tocause a parallel light to converge. The value of the optical path lengthdifference is 110 μm when F₂ =110 mm. Since the permissible value isΔΦ<(1/4)(λ2/Δλ)=30 μm, a blooming of the scanning beam is created.

The present embodiment provides a correction by eliminating the 80 μmdifference above. By ensuring that the focal distance of the sphericalconverging lens 116 is 10 mm, and disposing a semiconductor lens 114 ata distance of 11 mm from the lens, a converging wave having a focaldistance of 110 mm is obtained. Provided that the focal distance of themagnifying hologram lens 118 is 110 mm, the aforementioned optical pathlength difference can be eliminated by converting this converging waveinto a roughly parallel wave. As a result, a scanning beam free fromblooming is obtained.

FIG. 31 illustrates a seventh embodiment wherein a liquid crystalelement is used, as the first hologram 110, in place of an acousticoptic element used in the sixth embodiment. The magnifying hologram lens118 and the converging lens 116 are disposed between the light source114 and the first hologram 110.

Applying an electrical field to a transparent comb electrode having apitch of 0.5 μm generates a phase difference in the liquid crystal, thusforming a hologram. By changing the applied electrical field, afringe-free state and diffraction gratings having pitches 0.5 μm and 1μm are obtained. A light switching not affected by a wavelengthvariation, for example, can be realized by a semiconductor laser 114 forproviding an incident beam, which is diffracted by the second hologramplate 112. While FIG. 31 illustrates an embodiment where a single firsthologram is used, a plurality of such first holograms can also bealternatively disposed.

FIG. 32 illustrates an example where a reflection-type liquid crystal isused as the first hologram.

This example shows that the scanning beam can also be diffracted by afirst hologram manufactured from electro-optic crystals such as LiNbO₃,Sr_(x) Ba.sub.(1-x) NbO₃, KDP, GaAs, ZnO, and LiTaO₃ when used insteadof liquid crystal elements as employed in the seventh embodiment.Further, by using an acoustic optic element, a liquid crystal element,or an electro-optic crystal as the first hologram, the diffraction ofthe scanning beam in the first hologram can be electrically controlled.

With such electrical control, not only the beam scanning can beperformed at a higher velocity than with a mechanical means for movingthe first hologram, but also the downscaling of a light scanningapparatus is possible, and a mechanical performance degradation of thelight scanning apparatus is prevented because of a lack of any movableparts.

The specifics of the second and third modes of the light beam scanningapparatus of this invention will be described below, on the basis of thelight beam scanning apparatus shown in FIG. 23.

The first hologram 110 is of a flat-plate shape and converts aconverging wave into a diverging wave. The second hologram 112 is alsoof a flat-plate shape but converts a diverging wave into a convergingwave. The first hologram 110 is moved relative to the second hologram112. Designating the direction of the movement of the first hologram 110as the direction X, it is found from the foregoing analysis that a lightscanning apparatus free of displacement of the scanning beam positioncaused by a wavelength variation can be obtained by manufacturing thefirst and second holograms in such a way that the following equationsare fulfilled.

    k(δ)=k.sub.0 -(3/8) (1-k.sub.0.sup.2)(1-2k.sub.0)δ.sup.2

    k.sub.1 =-(3/8)(1-k.sub.0.sup.2) (1-2k.sub.0)δ.sup.2

    b.sub.0 =L/(1-2k.sub.0)

    a(δ)=L/k(δ)

    sin β=k(δ) sin δ

    W=L tan δ+b.sub.0 tan β

Assuming that a light is perpendicularly incident on the first hologram,a diffraction angle δ of the outgoing wave changes with time as thefirst hologram is moved.

FIG. 25(B) shows, using the parameter k₀, examples of specifications fordesigning the diffraction angle β of the second hologram and of thescanning distance W, when k₁, the desired value of b₀ and δ=35° aregiven.

FIG. 25(C) shows examples of configurations enabling approximately thesame image formation distance both at the outgoing angle δ of 0° and35°, and ensuring a flat image formation characteristic.

The equations below are employed in the above design.

    1/a(δ)=k(δ)L=(k.sub.0 +k.sub.1 δ.sup.2)/L

    sin β=k(δ) sin ##EQU14##

    W=L tan δ+b.sub.0 tan β

    b.sub.0 =L/(1-2k.sub.0)

In accordance with this invention, a light scanning apparatus free fromdisplacement of the scanning beam position, displacement of a focus ofthe scanning beam, or a blooming thereof, is obtained even with a lightsource having wavelength variation or wavelength dispersion. Further, alight beam scanning apparatus is obtained having not only the capabilityof correcting a wavelength variation but having also a flat imageformation characteristic. Since an inexpensive semiconductor laser or alight emitting diode can be used in a light beam scanning apparatus ofthis invention, low-cost manufacture of such light beam scanningapparatuses is possible. Since holograms can be mass produced, a lightbeam scanning apparatus less expensive than the conventional polygonscanner is obtained. This invention can be applied to a scanner for alaser printer, a POS scanner, a light head, a three-dimensional-shapeinspection apparatus, and a light switch.

The aforementioned first invention is characterized in that it providesuniform peripheral optical path lengths in a light flux, which opticalpath lengths are measured from the incident light to the image formationsurface. Such a configuration achieves correction for displacement of ascanning beam position.

Accordingly, as shown in FIGS. 33 and 34, when a plane wave is createdby a collimating lens 31 from a diverging light emitted from asemiconductor laser in a light source portion, and the parallel light isthen made to converge in the cross scanning direction by a cylindricallens 32 while the light is maintained so as to be a parallel light inthe scanning direction (or while the convergence is also effected in thescanning direction), the focus of the incident wave in the crossscanning direction should be brought onto the rotatable hologram 10.

The following problem may be expected to arise in the first invention.

1 The diffraction angle of the fixed hologram plate 20 needs to be assmall as about 0.5° in order to provide uniform peripheral optical pathlengths in the light flux, which optical paths are measured from thelight source to the image formation surface 4a, thereby leading to lowspatial frequency (the number of interference fringes per unit area), aninefficient diffraction, weaker light-power on the image formationsurface 4a.

2 In a so-called in-line hologram characterized by a small diffractionangle, the reference wave and the object wave need to be close to eachother when manufacturing a hologram, thus making the manufacturedifficult using a light exposure.

3 A small diffraction angle results in difficulty in separation ofhigh-order diffracted lights and it permits unnecessary light mixing.

In view of the above disadvantages, an object of the third invention isto provide a light beam scanning apparatus capable of allowing a largediffraction angle of the fixed diffraction grating even when uniformperipheral optical path lengths in the light flux, which optical pathlength are measured from the incident light to the image formation, areprovided.

FIG. 35 is a diagram illustrating the principle of the third invention.

The third invention comprises: a light source portion 201; a rotatablehologram 202; and a fixed plate 203 disposed between the rotatablehologram 202 and a scanning surface 204 and on which rotatable hologram202 a diffraction grating is recorded, wherein a light incident from thelight source portion is diffracted by a diffraction grating of therotatable hologram 202, a scanning is conducted with the diffractedlight by the rotation of the rotatable hologram 202, the same scanninglight is diffracted by the fixed plate 203 so as to conduct a lightscanning on the scanning surface 204, the invention characterized inthat the convergent position of the light incident on the rotatablehologram 202 is displaced from the surface of the rotatable hologrameither toward the image formation surface or toward the incident lightin a direction at right angles to the scanning direction, and in thatthe fixed plate 203 diffracts the diffracted light from the rotatablehologram 202, so that the peripheral optical path lengths are uniform,which optical path lengths are measured from the incident light to thescanning surface 204. Descriptions of the specific embodiments are givenbelow.

FIG. 36 is a diagram illustrating a configuration of a first embodimentof the third invention. In the figure, a diverging light from asemiconductor laser 210 is turned into a parallel light by a collimatinglens 211, and is caused to converge in the cross scanning direction Y bya cylindrical lens 212. In order to provide a fixed hologram plate 203with a diffraction angle under the condition that the one optical pathR₁ +R₂ +R₃ and the other optical path R_(1') +R_(2') +R_(3') areconfigured to be the same, the optical paths R₃ and R_(3') should havedifferent values. This same difference in values should exist betweenthe optical paths R₁ +R₂ and R_(1') +R_(2').

This embodiment is configured such that: the convergent position of thelight incident on the rotatable hologram 202 is in a direction away fromthe rotatable hologram face, towards the image formation surface in adirection at right angles to the scanning direction; a difference existsbetween the optical paths R₁ +R₂ and R_(1') +R_(2'), the fixed hologramplate 203 diffracts the diffracted light from the rotatable hologram 202to a large extent; and the peripheral optical path lengths in the lightflux are uniform. The optical paths are measured from the incident lightto the scanning surface 204a. Specifically, moving the cylindrical lens212 from its position in FIG. 34 nearer to the rotatable hologram 202enables the focal position to be set away from the rotatable hologram202 toward the image formation surface, in other words, at M₁ beyond therotatable hologram 202.

Accordingly, the incident wave in the form of a converging sphericalwave is diffracted by the rotatable hologram 202, is caused to convergealong the way, diverges, and is incident on the fixed hologram plate203.

Since the relationship R₁ <R_(1') stands valid regarding the opticalpaths R₁ and R_(1') measured from the rotatable hologram 202 to areference sphere surface, which sphere has a center M₁, and therelationship R₂ <R_(2') stands valid regarding the optical paths R₂ andR_(2') measured from the rotatable hologram 202 to the fixed hologramplate 203, the relationship R₃ <R_(3') stands valid regarding theoptical paths R₃ and R_(3') measured from the fixed hologram plate 203to a reference sphere surface, which sphere has a center M₂ (the imageformation surface 204a).

Accordingly, there arises a need to bend the scanning beam outgoing fromthe fixed hologram plate 203, thus ensuring a large diffraction angle ofthe fixed hologram plate 203, and achieving, instead of an in-line typehologram, an off-axis type hologram having a high diffractionefficiency.

Since the focal position M₁ is removed away from the rotatable hologram202 toward the image formation surface, the diffraction direction of thefixed hologram plate 203 needs to be a positive direction, that is, itneeds to be the same direction as the diffraction direction of therotatable hologram 202.

The conditions for eliminating displacement of the scanning beamposition are met, the displacement being due to a wavelength variation(variation of a center wavelength, multi-mode distribution variation)caused by variation in temperature of the semiconductor laser 210, whileat the same time the diffraction angle of the fixed hologram plate 203can be large, thus preventing a lowered light power, hence making themanufacturing of a hologram plate easy, and preventing the mixing ofunnecessary high-order diffracted waves.

FIG. 37 is a diagram describing the manufacture of a fixed hologramplate of the first embodiment of this invention.

As shown in the figure, the fixed hologram plate 203 is manufactured bythe interference of an object wave (spherical converging wave) and areference wave. Calling the phase of the object wave Φ_(O) and the phaseof the reference wave Φ_(R), the phase distribution Φ_(H) of a hologrammanufactured by the above object wave and the reference wave isrepresented by the equation (61) below.

    Φ.sub.H =Φ.sub.O -Φ.sub.R                      (61)

Since the diffracted wave of the rotatable hologram 202 is a cylindricalspherical wave, the phase Φ_(R) of the reference wave is expressed bythe equation (62) below, where the phase differences between thespherical wave having a point Z₀ as the center and the cylindrical waveare included. ##EQU15## where k₂ is the wave number of thereconstructing wave, X is a coordinate in the scanning direction, Y is acoordinate in the cross scanning direction, Z is a coordinate in adirection at right angles to the scanning direction and the crossscanning direction.

The object wave is a spherical aberration wave having a principal axis Aof the fixed hologram plate 203 as the center, which axis is hit by theoutgoing wave from the rotatable hologram 202 at the scanning center.This spherical aberration wave is a so-called "positive sphericalaberration wave", bending further toward the axis A as it travels alongthe axis A. The phase Φ_(O) of the object wave is given by the equation(63) below

    Φ.sub.O =k.sub.2 (C.sub.1 ·(X.sup.2 +Y.sup.2)+C.sub.0 ·Y)                                              (63)

where C₀ and C₁ are constants.

Accordingly, the distribution phase Φ_(H) of a hologram is expressed bythe equation (64) below. ##EQU16##

Accordingly, the directional cosines f_(x) and f_(y), below of theobject wave in the directions X (scanning) and Y (cross scanning) arethe partial derivatives of the distribution phase Φ_(O) of the hologramwith respect to X and Y, respectively and are thus given by theequations (65) and (66). ##EQU17##

Therefore, it is best to ensure that the optical paths (PQ) originatingin principal axes P₁, P₂, P₃ and P₄, and ending in points Q₁, Q₂, Q₃,and Q₄, namely the points on the fixed hologram plate 203 hit by thelight, are of the same length 1/2C₁.

It is possible to provide a diffraction angle in the Y (cross scanning)direction by the presence of the term C₀ ·y in the equation (63).

FIGS. 38 and 39 are diagrams showing the distribution of the scanningbeam intensity and illustrating the effect obtained in the firstembodiment of the third invention. FIG. 38 illustrates the scanning beamshape determined according to wave optics when the wavelength of thesemiconductor laser is of a single mode. FIG. 39 illustrates thescanning beam shape determined according to wave optics when thewavelength of the semiconductor laser is of a multi-mode.

Referring to FIG. 36, the distance between the rotatable hologram 202and the fixed hologram plate 203 is 223 mm, the distance between thefixed hologram plate 203 and the image formation surface 204a is 265 mm,and the optical axis distance between the rotatable hologram 202 and theconvergence point M₁ is 35.5 mm.

With such a configuration, the outgoing angle of the fixed hologramplate 203, which angle is provided so as to compensate for displacementof the semiconductor laser 210 incident beam due to the wavelengthvariation, is 14.4°, thus ensuring a large diffraction angle and an easyseparation of high-order lights.

When such an appropriate value was provided, an excellent linearscanning characteristic was obtained, where a straight-line scanningerror was less than ±0.1 mm and a linear scanning error less than 0.3%.

The scanning beam intensity distribution at the scanning center and thescanning distance of 146 mm when the wavelength of the semiconductorlaser 10 is of a single mode, are shown in FIGS. 8(A), (B), and (C).

As for cases where the wavelength of the semiconductor laser 210 is of amulti-mode, a case is considered, which case is affected by thewavelength variation, where the multi-mode width is as large as 2 nm andthe power ratio is 0.6 in contrast with the value of 1 taken at thewavelength center.

The actual semiconductor laser 210 is of a heavily centralized spectralcharacteristic.

The scanning beam intensity distributions shown in FIGS. 39 (A), (B),and (C) were obtained at the scanning center, at the scanning distanceof 73 mm, and at the scanning distance of 146 mm.

It is found from these results that, in a multi-mode distributionaffected by a wavelength variation, a scanning beam intensitydistribution not very different from the one of a single modedistribution is obtained, and that the effect of the semiconductor 210wavelength variation is completely compensated for.

Thus, it is possible to provide a large diffraction angle of the fixedhologram plate 203, prevent a reduction of light power on the imageformation surface, making the manufacturing of a hologram plate easy andpreventing the mixing of unnecessary high-order diffracted waves, whilefulfilling the conditions for eliminating displacement of the scanningbeam position due to a wavelength variation (variation of the centerfrequency, a distribution variation in multi-mode) of the semiconductorlaser 210.

FIG. 40 is a diagram showing a configuration of the second embodiment ofthis invention.

Referring to FIG. 40, configurations that correspond to theconfigurations shown in FIG. 36 are given the same reference notation asin the previous figure. This embodiment is configured such that, in thedirection at right angles to the scanning direction, the convergentposition of the light incident on the rotatable hologram 202 is nearerto the incident light than the rotatable hologram face, a difference isprovided between the optical paths R₁ +R₂ and the optical paths R_(1')+R_(2'), the diffracted light from the-rotatable hologram 202 isdiffracted to a great extent by the fixed hologram 203, so that theperipheral optical path lengths within the light flux, which opticalpaths are measured from the incident light to the scanning surface 204a,are uniform.

The diverging light of the semiconductor 201 is turned into parallellight with a collimating lens 211, and is caused to converge in thecross scanning direction Y by a cylindrical lens 212. By moving thiscylindrical lens 212 nearer to the semiconductor laser 210 than theposition shown in FIG. 34, the focal position can be set at M₁, that is,nearer to the incident light than the rotatable hologram 202.

Consequently, the converging spherical wave diverges after convergingalong the way, is diffracted by the rotatable hologram 202, and isincident on the fixed hologram plate 203.

The optical paths R₁ and R_(1') from the rotatable hologram 202 to thesurface of the reference sphere having the center M₁ have therelationship R₁ <R_(1'), while the optical paths R₂ and R_(2') from therotatable hologram 202 to the fixed hologram plate 203 have therelationship R₂ <R_(2'), so that the optical paths R₃ and R_(3') fromthe fixed hologram plate 203 to the surface of the reference spherehaving the center M₂ (the image formation surface 204) have therelationship R₃ <R_(3').

Accordingly, the scanning beam outgoing from the fixed hologram plate203 needs to be bent so as to allow the diffraction angle of the fixedhologram plate 203 to be large, thereby creating an off-axis typehologram instead of an in-line type hologram, and assuring a largediffraction efficiency.

Since the focal position M₁ is in the light source side of the rotatablehologram 202, the diffraction direction of the fixed hologram plate 203needs to be negative, that is, counter to the diffraction direction ofthe rotatable hologram 202.

Thus, the conditions for eliminating displacement of the scanning beamposition are met, the displacement being due to a wavelength variation(variation of a center wavelength, or multi-mode distribution variation)caused by variation in temperature of the semiconductor laser 210, whileat the same time the diffraction angle of the fixed hologram plate 203can be large, thus preventing a lowered light power, making themanufacturing of a hologram plate easy, and preventing the mixing ofunnecessary high-order diffracted wave.

Other than the above-described embodiments, this invention allows thevariations listed below.

1 As shown in a description of an electrophotograph printing apparatuswhere a light beam scanning apparatus is applied, this invention can beapplied to such apparatuses as a laser drawing apparatus and a laserinspection apparatus.

2 Although the use of a hologram as a diffraction grating was assumed, adiffraction grating can be configured in other ways.

3 Although a disk shaped rotatable hologram was assumed in thedescription, a drum shaped rotatable hologram can also be employed.

4 Although the movement of the convergence position was described inaccordance with the movement of the position of the cylindrical lens212, the movement of the focal distance of the cylindrical lens 212 canalso be utilized.

This invention is not limited to the embodiments described so far, andvariations are possible within the scope of this invention and are notexcluded from the scope of this invention.

As has been described, the third invention ensures that the convergenceposition of the light incident on the rotatable hologram 202 is nearer(in a direction at right angles to the scanning direction) to the lightsource than the surface of the rotatable hologram, that a difference isprovided between the optical paths R₁ +R₂ and R_(1') +R_(2'), and thatthe diffracted light from the rotatable hologram 202 is diffracted to agreat extent by the fixed hologram plate 203 so that the peripheraloptical paths within the light flux, which optical paths originate inthe incident light and end in the scanning surface 204, have uniformlengths. Thus, it is possible to provide a large diffraction angle ofthe fixed hologram 203 and to prevent a light-power reduction on theimage formation surface, while fulfilling the conditions for eliminatingdisplacement of the scanning beam position, displacement being due to awavelength variation caused by temperature variation of thesemiconductor laser 210.

Because it is possible to provide a large diffraction angle of the fixedhologram 203, a diffraction grating can be manufactured easily, makingthe provision thereof inexpensive and stable. The large diffractionangle of the fixed hologram 203 also means that a mixture of unnecessaryhigh-order diffracted waves can be prevented.

When applying a light beam scanning apparatus to such apparatus as alaser printer, a scanning beam is required to scan on a photoconductivedrum, always at a constant velocity. Therefore, a fixed hologram plateis equipped with a correction function for ensuring linearity. As shownin FIG. 41, the linearity correction function of the fixed hologramplate in the first invention is such that considering that the scanningvelocity of the light beam used in the constant angular velocityscanning by the rotatable hologram is greater toward the scanning endthan at the center, as shown by a solid line, a diffraction angle islarger toward the scanning end than at the center, as shown in FIG. 33,so that the diffraction toward the center takes place and the scanningvelocity at each scanning position remains a constant velocity V₀ (seeFIG. 41(A)).

Therefore, as shown in FIGS. 41(B) and (C), the object wave forproducing interference fringes of the fixed hologram plate 20 needs tobe a spherical aberration wave. Further it is required that the amountof aberration of the object wave is maintained at the same level in boththe scanning direction and the cross scanning direction, in order toobtain the same beam radius on the image formation surface 4 in both thescanning direction and the cross scanning direction.

However, the following problem may be expected to arise in the firstinvention.

In order to achieve downsizing of an apparatus, the optical path fromthe rotatable hologram 10 to the image formation surface 4 should beshort (see FIG. 1), and the optical path from the rotatable hologram 10to the fixed hologram plate 10 is preferably short.

Since maintaining the scanning width of the rotatable hologram 10 causesthe scanning width on the image formation surface 4 to be small, thescanning width of the rotatable hologram 10 needs to be large. Thisenlarges the angle incident on the fixed hologram plate 20, causing thediffracted light to bend too much toward the center given the samespatial frequency, with the result that the linearity exhibits adegradation, as shown by broken lines in FIG. 41(A).

It is an object of this fourth invention to provide a light beamscanning apparatus that does not exhibit degradation in linearity, andwhich is free from beam aberration even when the optical path isshortened.

FIG. 42 is a diagram depicting a configuration of an embodiment of thisinvention (cross scanning direction). FIG. 43 is a diagram depicting aconfiguration of an embodiment of this invention (scanning direction).FIG. 44 is a diagram describing a fixed hologram plate of an embodimentof this invention.

Viewed along the cross scanning direction of FIG. 42: the diverginglight of the semiconductor laser 310 is turned into plane wave by acollimating lens 311; is caused to converge on the rotatable hologram302 in the cross scanning direction Y, convergence being effected by aY-side cylindrical lens 313 and via an X-side cylindrical lens 312; isdiffracted by the rotatable hologram 302; is incident on the fixedhologram plate 303; is diffracted and made to converge again; and isfinally convergent on an image formation surface 304a on aphotoconductive drum 304.

On the other hand, viewed in the scanning direction of FIG. 43; thediverging light of the semiconductor laser 310 is turned into plane waveby a collimating lens 311; is made to converge by an X-side cylindricallens 312; is further made to converge, via the Y-side cylindrical lens313, at M1 beyond the rotatable hologram 302 in the scanning directionX; is diffracted by the rotatable hologram 302 for scanning; is incidenton the fixed hologram plate 303; is diffracted and made to convergeagain; and is finally convergent on the image formation surface 304a onthe photoconductive drum 304.

This fixed hologram plate 303 allows different rates of variation ofdirection in cosines of the object wave in the scanning direction andthe cross scanning direction.

Thus, while the object wave for creating interference fringes of thefixed hologram plate 303 is conventionally an isotropic spherical wave,this invention allows for different amounts of aberration in thescanning direction and the cross scanning direction, as shown in FIGS.44(A) and (B), and a linear scanning may be achieved for any point ineach direction because aberration is provided such that the distancefrom the optical axis to the hologram face is uniform.

On the other hand, in the conventional technology, shortening theoptical path between the rotatable hologram 302 and the fixed hologramplate 303, and enlarging the scanning width of the rotatable hologram302 causes a degradation in the linearity because high spatial frequencyof the fixed hologram plate 303 brings the diffracted light toward thescanning center.

In order to lower the spatial frequency in the scanning direction, thelength F1, the distance between the optical axis and the hologram face,is longer than the regular distance F2 in the cross direction shown inFIG. 44(A). F1 is the same at any point at which an amount of aberrationin the main scanning direction in FIG. 44(B) is determined.

That is, the distances between the optical axis and the hologram face inthe cross scanning direction fulfill the equation (71) below.

    P.sub.1 Q.sub.1 =P.sub.2 Q.sub.2 =P.sub.3 Q.sub.3 . . . =F2(71)

Further, the distances between the optical axis and the hologram face inthe scanning direction fulfill the equation (72) below.

    P.sub.1' Q.sub.1' =P.sub.2' Q.sub.2' =P.sub.3' Q.sub.3'  . . . =F1(72)

The distances F2 and F1 are arranged so that a condition represented bythe inequality (73) below stands valid.

    F1>F2                                                      (73)

This arrangement ensures that the spatial frequency of the fixedhologram plate 303 in the scanning direction is low, and that anover-correction of the linearity is prevented.

The phase Φ_(O) (X, Y) of the object wave fulfilling the above equation,the wave being recorded on the fixed hologram plate 303, is given by theequation (74) below. ##EQU18## where X and Y are coordinates in thescanning direction and the cross scanning direction respectively, theorigin of the coordinates being the scanning center of the fixedhologram plate 303, and k₂ is a wave number derived from the wavelengthλ₂ of the reconstructing wave.

As can be seen from the equation (74), since C₀ ·Y is a correction termprovided because of an off axis incidence, the phase of the object waveneeds to be an ellipse as shown in FIG. 44(D), which is different from acircle shown in FIG. 44(C) produced by the conventional isotropicspherical wave.

The phase Φ_(R) of the reference wave in this case is represented by thephase difference between a spherical wave having a center Z₀ and acylindrical wave, and is expressed by the equation (75) below: ##EQU19##where Z is a coordinate in a direction at right angles to the scanningdirection and the cross scanning direction.

Accordingly, a direction cosine f_(x) in the scanning direction of theobject wave and a direction cosine f_(y) in the cross scanning directionare the results of partial differentiation of the equation (74), and aregiven by the following equations (76) and (77). ##EQU20## When theequations (76) and (77) are compared, it is found that the rate ofvariation of the direction cosine f_(x) of the object wave in thescanning direction is smaller than the rate of variation of thedirection cosine f_(y) of the object wave in the cross scanningdirection, which makes it possible to make small the spatial frequencyf_(x) in the scanning direction.

Changing the distance to F1 produces a difference in the beamimage-formation distances in the scanning direction and the crossscanning direction on the image formation surface 304a, therebynecessitating the correction thereof.

As shown in FIG. 43, assuming that the image formation at the imageformation distance L takes place in the cross scanning direction whenthe fixed hologram plate 303 is of a parameter F2, the image formationin the scanning direction at the image formation distance L, when theparameter is F1, requires that the focal distance d₁ of the divergingwave outgoing from the rotatable hologram 302 be obtained by a knownequation (78) shown below.

    1/d.sub.1 =1/F1-1/L                                        (78)

Since the position of the rotatable hologram 302 cannot be changedbecause of the image formation condition in the cross scanningdirection, the incident focal distance d₀ of the incident wave isadjusted.

That is, the beam radius difference between the scanning direction andthe cross scanning direction can be corrected by controlling the focaldistance d₀ of the incident light by means of the X-side cylindricallens 312, setting the incident convergence point M₁ to be beyond therotatable hologram 302 so that the outgoing wave originating position M₂of the rotatable hologram 302 is removed from the fixed hologram plate303 by a distance d₁ and the image formation takes place at the imageformation distance L in the scanning direction at the parameter F1.Normally, the X-side cylindrical lens 312 can be omitted and thecontrolling in the direction X can be performed with the position LDcontrol by means of the collimating lens.

Specifically, this embodiment is configured such that the optical pathL₁ between the rotatable hologram 302 and the fixed hologram plate 303is as short as 234 mm, the optical path L between the fixed hologramplate 303 and the image formation surface 304a is 276 mm, and F1 and F2are 279 mm.

With this configuration, a beam having only a 50 micron deviation fromthe straight line and little aberration can be obtained by choosing anappropriate parameter, a linearity, however, is as bad as at a -1.4%level.

When a step was then taken to make F1 become larger than F2, F1 being356 mm and F2 being 226 mm, L₁ became 273 mm and L₂ became 226 mm.Although the overall optical paths remained unchanged, a beam having a50 micron deviation from the straight line, no aberration, and alinearity as good as a 0.6% level was obtained. These values aresufficient for actual operation.

In the equation (74), the constant k₂ used in determining the phase ofthe object wave is equal to wave number derived from the wavelength λ₂of the reconstructing wave, and this constant can be different from thewave number of the reconstructing wave.

In case the semiconductor laser 310 is used for producing areconstructing wave, the wavelength λ₂ is 780 nm. Since the wavelengthsensitivity of a hologram material having a high diffraction efficiencybelongs to a wavelength range shorter than this wavelength, manufactureof a hologram should be done with a laser light source having a shorterwavelength, for example, an argon laser having a wavelength λ₁ of 488 nmmay be used.

Thus, because the wavelength of the object wave is λ₁ and the wavelengthof the reconstructing wave is λ₂, it is required to change the phase ofthe object wave.

Given that the wave number derived from the wavelength λ₁ of theconstructing wave is k₁, and that the wavelength ratio between theconstructing wave and reconstructing wave is S (=λ₂ /λ₁), the phaseΦ_(O) (X, Y) of the object wave is given by the equation (79) below.##EQU21##

Accordingly, the distances F1 and F2 being of the object wave aredetermined by multiplying, F1 and F2 chosen in accordance with theequation (74), by a wavelength ratio, when the wave number derived fromthe wavelength λ₁ of the hologram constructing wave is k₁, whichconstructing wave is not produced by the semiconductor laser 310.

Thus, even when the distance from the hologram rotatable hologram 302 tothe image formation surface 304a is shortened for the purpose ofdownsizing, a linear scanning and a light scanning free from light-beamaberration can be achieved.

This invention is not limited to the above-mentioned embodiment and thefollowing variations are also possible.

1 Although the application of a light beam scanning apparatus to anelectrophotograph printing apparatus was assumed, it can also be appliedto a laser drawing apparatus or a laser inspection apparatus, forexample.

2 Although a diffraction grating in the form of a hologram was assumed,the diffraction grating can also be of another configuration.

3 Although a rotatable hologram of a disk-shape was assumed, othershapes including a drum-shape can also be applicable.

4 While, in the aforementioned example, degradation of linearity in thenegative direction was assumed, F1 should be made smaller than F2 andthe rate of variation of the spatial frequency in the scanning directionshould be larger than the rate of variation of the spatial frequency inthe cross scanning direction, when the degradation in the positivedirection takes place.

As has been described, this invention has the following effects.

1 Since the rate of variation of the direction cosine of the object wavein the scanning direction of the fixed hologram plate 303 is configuredto be different from the rate of variation of the direction cosine ofthe object wave in the cross scanning direction, a linear scanning isachieved even when optical path lengths are made shorter.

2 Even when the beam image formation distance varies in the scanningdirection and the cross scanning direction, the variation of the amountof the beam aberration can be corrected by configuring the lightincident on the rotatable hologram 302 such that the light has differentfocal distances in the scanning direction and the cross scanningdirection, so that there is no beam aberration.

An object of this invention is also to provide a light beam scanningapparatus exhibiting no degradation in linearity even when the opticalpath length are configured to be short, the scanning apparatus beingfree from light beam aberration.

FIGS. 45 and 46 are diagrams illustrating an embodiment of the fifthinvention. Referring to FIG. 45, a focal distance F needs to be large inorder to reduce distortion, as is known in a convex lens optical system.To achieve the same with a hologram, the distance from the point lightsource of the object wave to the hologram face needs to be longer. Thatis, referring to the figure, the position of the point light source ofthe spherical wave constituting the object wave should be at D_(0')instead of at D₀.

Providing a larger distance F of the reference spherical wave puts theimage formation position in the center further away from the lightsource, so that a distortion, though decreased to some extent, cannot becompletely eliminated.

The phase Φ_(R') (X, Y) of the reference wave shown in theabove-mentioned equation (62) is expressed by the equation (81) below.##EQU22##

That is, the phase of the reference wave is determined from thespherical wave phase represented by the first term in the equation (81)and the phase difference of the cylindrical wave represented by thesecond term. In the scanning direction, the reference wave is a roughlyparallel perpendicular light. k₂ is the wave number (2π/λ₂) derived fromthe wavelength λ₂ of the reconstructing wave, and is the same as thewave number k₁ derived from the wavelength λ₁ of the constructing wave,and contains the wavefront having a spherical wave optical axis D₀ O, D₀O being the same as the distance between the hologram optical axis andthe face of hologram disk 2.

Accordingly, the manufacture, using a hologram, of a non-spherical lens,where the focal distance is F(D₀ O) in the center and the focal distanceis greater toward the end of the lens, is required. The phase Φ_(R) ofthe reference wave for manufacturing the hologram is given by theequation (82) below. ##EQU23##

The difference between the equations (81) and (82) is that the wavenumber k₁, derived from the wavelength λ₁ of the constructing wave, isused instead of the wave number k₂ derived from the wavelength λ₂ of thereconstructing wave.

If this wavelength λ₁ is greater than λ₂, the wave number k₁ is smallerthan k₂ and the phase of the reference wave becomes smaller.

Consequently, a wavefront different from the wavefront manifested whenthe diffracted wave from the rotatable hologram 402 is incident on thefixed hologram plate 403, is recorded on the fixed hologram plate 403.

In this way, a non-spherical lens, where the focal distance is F(D₀ O)at the center and the focal distance is greater toward the end of thelens, can be realized with a hologram on the condition that thereconstructing-wave wavelength λ₂ is employed during reconstruction.

The above arrangement also enables the obtaining of a scanning beam freeof distortion, in which beam the convergence position in the scanningcenter remains unchanged and the convergence position at the scanningend is shifted toward the image formation surface 404a. The arrangementalso enables a linear scanning, because the beam at the scanning end isshifted toward the outside as shown in FIG. 46(B).

While the coefficient k₁ is used in the term relative to the cylindricalwave in the phase equation (82), it is possible, in order to obtain thesame function as above, to retain the coefficient k₂ in the termrelative to the cylindrical wave and employ the phase equation (83)below, because only a spherical wave is responsive to theabove-mentioned aberration correction and linear scanning operation. Thesame effect is achieved by ensuring that the wavefront of the referencewave is different from the wavefront of the wave incident from therotatable hologram 402 on the fixed hologram plate 403. The phase of thereference wave in this case can be obtained by changing the point lightsource position D₀ of the reference wave, or specifically, bysubstituting Z₀ for Z_(0') in the equation (82). Thus the followingequation holds. ##EQU24##

For example, when the fixed hologram plate 403 manufactured inaccordance with the phase equation (81) is employed while providing anoptical path length of 500 mm, a beam aberration radius is as large as80 microns, a linearity is below the 0.5% level, and the deviation fromthe straight-line is controlled to be less than 50 microns.

When employing the fixed hologram plate 403 manufactured according tothe phase equation (82), the beam aberration radius is reduced to 20microns, the linearity is improved to a level below 0.4%, and thedeviation from a straight line is maintained so as to be below a 50micron level.

The wavelength ratio (λ₁ /λ₂) in this case is set to be at 1.02.

Thus, once a necessary phase is obtained, a hologram can be manufacturedby drawing a pattern with an electron beam or a laser, assisted in somecases by an auxiliary optical system. Extracting this hologram patternenables the reconstruction of the fixed hologram plate 403.

This way, even when the optical path length is short, a distortion canbe eliminated and the linearity can be improved by changing the phase ofthe reference wave for manufacturing interference fringes of the fixedhologram plate to obtain a non-spherical lens hologram.

This invention is not limited to the above embodiments but the followingvariations are also possible.

1 Although an electrophotograph printing apparatus was assumed as anapparatus in which to apply the light beam scanning apparatus, the lightbeam scanning apparatus can also be applied to other apparatuses, suchas a laser drawing apparatus or a laser inspection apparatus.

2 Although a disk-shape rotatable hologram was assumed, other shapesincluding a drum-shape can also be applicable.

3 The point-light source position of the reference wave, spherical wave,and cylindrical wave can also be shifted in the Y-axis direction (crossscanning direction).

As has been described, this invention allows the fixed hologram plate tohave an interference fringe distribution produced by the wave, where thewave number of the spherical wave has a phase different from the wavenumber derived from the wavelength of the reconstructing wave, therebyallowing the construction of a non-spherical lens and a scanning beamfree of aberration or distortion, while enabling the reduction of theoptical path length. Also, because it is a non-spherical lens, a linearscanning is realized even when the optical path length is configured tobe short.

The above-mentioned first through fifth inventions are equipped with arotatable hologram and a fixed hologram plate. While methods ofmanufacturing interference fringes configuring a hologram have beendescribed in detail in the above-mentioned inventions, optimumconditions for the shapes of a rotatable hologram and a fixed hologramplate have not been considered at all.

Generally, the length of a fixed hologram plate in the scanningdirection is set to be smaller than the scanning distance of a lightbeam. This is because it is known that the smaller a fixed hologramplate, the easier its manufacture.

Experiments were carried out to determine various characteristics.Attention was paid to the length of a fixed hologram plate and ascanning distance of a light beam. Two cases were considered, namely, 1a case where the scanning distance of a light beam is longer than thelength of a fixed hologram plate, and 2 a case where the scanningdistance of a light beam is shorter than the length of a fixed hologramplate. The results of the above mentioned experiments are shown in FIGS.47 and 48. FIG. 47 shows results in a case 1 where the scanning distanceof a light beam is longer than the length of a fixed hologram plate,while FIG. 48 shows results in a case 2 where the scanning distance of alight beam is shorter than the length of a fixed hologram plate.

In the experiments, the basic configuration shown in FIG. 1 was employedas the configuration of a light beam scanning apparatus. The distanceseparating the rotatable hologram 10 and the fixed hologram plate 20 isset to be 275 mm. The distance separating the fixed hologram plate 20and the photoconductive drum 3 is 391 mm. The scanning width therein is291 mm. In the case 1 experiment, the length of the fixed hologram plate20 in the scanning direction is set to be 244 mm (shorter than thescanning width 291 mm), while in the case 2 experiment, the length ofthe fixed hologram plate 20 in the scanning direction is set to be 344mm (longer than the scanning width 291 mm). Both were optimally designedusing a computer.

The results in FIG. 47 show that the linearity and the beam aberrationare greater when the scanning distance of a light beam is longer thanthe length of the fixed hologram plate 20, and that a successful lightbeam scanning cannot be performed in this case.

On the other hand, the results in FIG. 48 show that the linearity, thebeam aberration, and the displacement of position in the scanningdirection due to a wavelength variation are small enough for applicationin a laser printer, for example, when the scanning distance of a lightbeam is smaller than the length of the fixed hologram plate 20 (in otherwords, when the fixed hologram plate 20 is shorter than the scanningdistance of a light beam).

Accordingly, as shown in FIG. 49, a light beam scanning in which anexcellent linearity is obtained, and affected by little beam aberrationand displacement of beam position, is achieved by setting the length X₁of the fixed hologram plate 20 measured in the scanning direction, to beshorter than the scanning distance X₂ of a light beam.

The light beam scanning apparatuses in the above-described first, andthird through sixth embodiments are configured such that a rotatablehologram equipped with a plurality of hologram lenses on thecircumference of a circle is rotated with linearity, a laser light isincident on the rotatable hologram via a collimating lens, and the laserlight diffracted thereby is put through the fixed hologram plate so thatan image is formed on an image formation surface.

However, in this light beam scanning apparatus consisting of a rotatablehologram and a hologram optical system, a motor is required to rotate arotatable hologram. This motor has a disadvantage in that it isexpensive and there is an upper limit to its revolution speed (10,000rpm with a normal bearing; 40,000 to 50,000 rpm with an air bearing).While a hologram optical system has an advantage in that it is costeffective, being less expensive than an Fθ lens optical system, it has adisadvantage when it comes to downsizing and increasing the velocitythereof.

A light scanning apparatus employing a galvanomirror is characterized inthat, because it uses a sine wave oscillation, the scanning frequency(20 kHz, for example) is markedly higher than the motor rotationfrequency (10,000 rpm=167 Hz, for example). However, since the mirroroscillation is of a sine mode, a difference results in the scanningvelocity at the center and the periphery of the scanning surface, thusmaking it difficult to achieve a linear scanning (a scanning where lightscanning velocities are the same at the center and at the ends). Amethod has already been developed where a saw-tooth waveform is used asa driving wave of a galvanomirror and a linear range of an oscillationmode is enlarged so that a linear scanning is secured. However, thismethod entails disadvantages in that an oscillation frequency becomeslow, and the scanning velocity becomes low (several hundred Hz) ascompared with the mirror rotation. Moreover the need for adding agalvanowaveform driving circuit raises the cost.

An object of this invention is to provide a small and inexpensive lightbeam scanning apparatus capable of performing a high-speed linearscanning.

FIGS. 50(A) and (B) are, respectively, a configuration diagram and a topview of the first embodiment of this invention.

Referring to FIG. 50(A), a laser light outgoing from the laser diode 530is turned into parallel light by a collimating lens 53 and is incidenton a galvanomirror 532.

A galvanomirror 532 is driven by a sine driving waveform generated in asine-wave mode driving circuit 533, and produces a sine wave modeoscillation as shown in FIG. 51(A). This oscillation is done at afrequency of 20 kHz, for example. This configuration is not enough toachieve a linear scanning.

As shown in FIG. 50(B), the laser light reflected by the above-mentionedgalvanomirror 532 forms an image on an image formation surface 536 via afirst hologram 534 and a second hologram 535 disposed on parallelplanes.

As shown in FIG. 52(A), the first hologram 534 is of a fringe patternsuch that a fringe density at the center is about 1700 fringes/mm and afringe density at the ends is about 1800 fringes/mm, and such that areverse sine conversion is thereby performed, where a diffraction angleis gradually greater at both ends than at the center. The first hologramhas a light scanning velocity conversion characteristic shown in FIG.51(B). As shown in FIG. 51(C), this light scanning exhibits, by beingallowed to go through the first hologram 534, a velocity characteristicwhere the velocity remains at the same level at the center of the imageformation surface and drops at the ends.

The second hologram 535 is of a fringe pattern such that a fringedensity at the center is about 400 fringes/mm and a fringe density atthe ends is about 700 fringes/mm, and such that a tangent conversion isthereby conducted, where a diffraction angle remains uniform from thecenter to the left and right middle portions and increases steeply atthe left and right ends. The second hologram has a light scanningvelocity conversion characteristic shown in FIG. 51(D). The lightoutwardly diffracted by the first hologram 534 is inwardly diffracted,by being allowed to go through the second hologram 535, and, as shown inFIG. 51(E), the light scanning velocity having a characteristic shown inFIG. 51(C) is corrected such that the velocity is constant in a rangeextending from the center and covering more than half the imageformation surface. A linear scanning is possible with a sine wave modedriving by using the first hologram 534 and the second hologram 535.

FIG. 53 illustrates a configuration of the second embodiment of thisinvention. In the figure, parts that are the same as those in FIG. 50are given the same reference notations as in the previous figure.

Referring to FIG. 53, a laser light passing through a collimating lens53 is incident on a torsion bar mirror 540. The torsion bar mirror 540is driven by a sine-mode driving circuit 533, and produces a sine wavemode oscillation. The laser light reflected by the torsion bar mirror540 forms an image on an image formation face 536 via a first hologram542 and a second hologram 543 disposed as shown in a side view in FIG.54(A) and a top view in FIG. 54(B).

As shown in FIG. 55(A), the first hologram 534 allows a largerdiffraction angle toward the left and right ends than at the center, sothat a reverse sine conversion can be performed. The fringe patternthereof is of an arc shape with its center residing on a line extendingabove the center portion of the hologram, and maintains the sameperpendicular upward diffraction angle over the entire longitudinaldirection range.

As shown in FIG. 55(B), the second hologram 543 is of a fringe patternsuch that a diffraction angle remains the same from the center portionto the left and right middle portions, and a diffraction angle isincreased steeply at the left and right ends, so that a tangentconversion can be performed. Also, the fringe pattern thereof is of anarc shape with its center residing on a line extending below the centerportion of the hologram, and maintains the same perpendicular downwarddiffraction angle over the entire longitudinal direction range.

Thus, an oscillation frequency, that is a scanning frequency, becomes 10to 100 times higher by using a sine-wave mode signal mirror. In case ofa motor rotation, a rotation frequency has an upper limit of 1 kHz(50,000 rpm) even when an expensive air bearing is used. When a polygonmirror is used as a scanning means, such a rapid revolution may cause abrittle fracture, thus prohibiting the use of a glass-base mirror. It isknown, however, that sine wave mode oscillation mirrors having arotation frequency of 20 kHz are generally available, and can actuallybe used in a high-frequency driving. A polygon mirror is not onlyexpensive itself but has a disadvantage in that an expensivehigh-precision motor is required (a rotation jitter of below 0.1% isrequired), making the composite large and heavy. By comparison, agalvanomirror and a torsion bar mirror are known to be small andinexpensive (with an exception that a saw-tooth wave oscillation mirroris expensive). As for resolution, 400 dpi (400 dots/inch) can be easilyachieved by using a hologram or an optical system.

In the embodiment shown in FIG. 50, a fringe interval is greater in thecenter portion of the hologram 534 and in the end portions of thehologram 535, thus increasing the ratio of non-diffracted light, andpossibly creating a light-amount variation in each portion of the imageformation surface 536. In the embodiment shown in FIG. 53, the ratio ofnon-diffracted light is very small in each portion of the holograms 542and 543, thus not creating a light-amount variation in each portion ofthe image formation surface 536.

Further, since the above embodiment employs a hologram optical systemwhere the first holograms 534 and 542 and the second holograms 535 and543 are combined, even when a minute change occurs in the diffractiondirection of the laser light from each hologram, the change being due toa laser light wavelength-variation caused by a fluctuation intemperature of the laser diode 530, the changes in the diffractiondirections of the first and second holograms are absorbed by each other,so that the variation of the beam image-formation position can beprevented.

As described above, a light scanning apparatus of this invention isextremely useful in that it allows a high-speed linear scan, and allowsconfiguration of a small-scale inexpensive apparatus.

This invention relates to an optical divider employing two hologramplates.

Recent speed-up of a VLSI circuit (Very Large Scale Integrated Circuit)is bringing about an increase in speed of a clock supplied in a VLSIcircuit. As the degree of integration grows, it is desired that a clocksignal be shared in VLSI circuits.

A method of supplying a clock by means of leads generates a delay in aclock signal due to lead capacitance, thereby obstructing thesynchronization among each VLSI circuit.

To solve this problem, a method has been proposed for supplying a clockto a VLSI by means of a light signal. In a configuration in whichsynchronization is achieved with a light signal, synchronization iseasily achieved because of the lack of delay such as that found in aconfiguration using leads. A semiconductor laser, which has an advantageof compactness, is generally used as a light source for this lightsignal, a laser light from the semiconductor laser being divided by anoptical divider and being incident on a photodetector provided in eachVLSI.

It may be expected that, when a wavelength variation occurs in thissemiconductor laser, the position of light converging on thephotodetector provided in each VLSI circuit is displaced, and that ablooming is caused, thus preventing accurate synchronization.

An object of the eighth embodiment is to provide an optical element inwhich a displacement of a converging light position, and an occurrenceof blooming are prevented.

FIGS. 56 and 57 are diagrams illustrating a configuration of opticalelements constituting an embodiment of this invention.

In the figures, 600 represents a first hologram, and 601 a secondhologram. The first hologram 600 and the second hologram 601 aredisposed opposite to each other and are separated by a certain distance.Below the second hologram 601 are disposed photodetectors 602 in amatrix (6×6 matrix in this embodiment), the photodetectors beingprovided in VLSI circuits.

A laser light emitted from a semiconductor laser not shown in the figureis incident on the first hologram 600 and is divided thereby into lightsof equal intensity. A Damann grating, for example, is available as thefirst hologram 600. Ideally, the incident light is a converging lightbut it can also be a parallel light.

The diffracted light, after being divided into lights of equal intensityby the first hologram 600, is incident on the second hologram 601. Thesecond hologram 601 is constructed in a matrix (6×6 matrix)corresponding to the photodetectors 602, also disposed in a matrix, andthe above-mentioned diffracted light is uniformly made to converge oneach of the photodetectors 602 by the second hologram 601. In case aparallel light is outgoing from the first hologram 600, it is best forthe second hologram 601 to be of a phase derived from expanded paraxialphase shift. Supposing that this phase is Φ(X, Y), adjusting Φ(X, Y) tobe the value obtained by the following equation to Φ(X, Y) ensures thatthe laser light converges on each photodetector element:

    Φ(X, Y)=k.sub.2 ×(X.sup.2 +Y.sup.2)/2F,

where

X, Y: coordinates of each photodetector

k₂ : constant

F: distance from the first hologram to the photodetector.

As shown in FIG. 57, assuming that the sum of three optical paths: theoptical path R₁ of the light incident on a photodetector elementbelonging to a matrix of photodetectors 602 and located at a point n;the optical path R₂ (n) of the outgoing wave corresponding to the pointn; and the optical path R₃ (n) of the diffracted light corresponding tothe point n; is E₀, and that the optical path of the principal axispoint of the light flux incident on the point n is E₁ ; and that thedifference thereof (E₀ -E₁) is δW_(m) (n), E is defined from theequation below. ##EQU25## By carrying out an optimization so that Ebecomes minimized, the variation of a convergence point due to awavelength variation of the semiconductor laser is minimized. Thisminimization ensures that lights of uniform intensity can be distributedamong the photodetector elements constituting the photodetectors 602,and that an occurrence of displacement of a convergence point and of ablooming is prevented.

FIG. 58 illustrates an example where two holograms are provided not inan in-line configuration but with the axes thereof being displaced withrespect to each other, in order to increase the efficiency of lightusage. FIG. 59 illustrates an example where a liquid crystal valve 605is provided between the second hologram 601 and the photodetectors 602,so that a laser light can selectively irradiate a plurality ofphotodetector elements constituting the photodetectors 602.

An object of the ninth embodiment is to provide a light beam scanningapparatus exhibiting no degradation in linearity and no beam aberrationeven when the optical path lengths are configured to be short. Anotherobject of this invention is to provide a method of manufacturing a fixedhologram plate of a light beam scanning apparatus exhibiting nodegradation in linearity and no beam aberration even when the opticalpath lengths are short.

FIG. 60 is a diagram illustrating the principle of this invention. Thisinvention provides a light beam scanning apparatus wherein an incidentlight from a light source portion 701 is diffracted by a rotatablehologram 702, a scanning being conducted by the rotation of thisrotatable hologram 702 using the diffracted light, the scanning lightbeing diffracted by a fixed hologram plate 703 so as to carry out ascanning on a scanning surface 704.

The apparatus is characterized in that the fixed hologram plate 703 hasa phase distribution Φ_(H) determined from the equation (91) below,given that the scanning direction is X and the cross scanning directionis Y; and in that the light incident on the above-mentioned rotatablehologram 702 is configured to have different focal distances in thescanning direction and the cross scanning direction. ##EQU26##

Also, this invention provides a light beam scanning apparatus, whereinan incident light from the light source portion 701 is diffracted by therotatable hologram 702, a scanning being conducted by the rotation ofthe rotatable hologram 702 using the diffracted light, and the scanninglight is diffracted by the fixed hologram plate 703 so as to carry out ascanning on the scanning surface 704.

This apparatus is characterized in that the above-mentioned fixedhologram plate 703 has a phase distribution Φ_(H) determined from theequation (92) below, given that the scanning direction is X and thesecond scanning direction is Y, and in that the light incident on therotatable hologram 702 is configured to have different focal distancesin the scanning direction and the cross scanning direction. ##EQU27##

Also, this invention provides a method of manufacturing the fixedhologram plate 703 of a light beam scanning apparatus, wherein anincident light from the light source portion 701 is diffracted by therotatable hologram 702, a scanning being conducted by the rotation ofthe rotatable hologram 702 using the diffracted light, and wherein thescanning light is diffracted by the fixed hologram plate 703 so as tocarry out a scanning on the scanning surface 704.

This method is characterized in that an interference fringe distributionof the fixed hologram plate 703 is created on the basis of a wave havinga spherical aberration, the aberration amount being measured in the Ydirection, astigmatism, and coma; and a wave having an aberration amountmeasured in the X direction, and has a spherical aberration andastigmatism of a wavelength different from that of a reconstructingwave.

Further, this invention provides a method of manufacturing the fixedhologram plate 703 of a light beam scanning apparatus, wherein anincident light from the light source portion 701 is diffracted by therotatable hologram 702, a scanning being conducted by the rotation ofthe rotatable hologram 702 using the diffracted light, and wherein thescanning light is diffracted by the fixed hologram plate 703 so as tocarry out a scanning on the scanning surface 704.

This method is characterized in that an interference fringe distributionof the fixed hologram plate 703 is created on the basis of two waves,one wave having a spherical aberration, the wave coming from a pointlight-source and a coma, and being an origin of the Y directioncomponent of an elliptic phase wave, and the other wave being an originof a line light-source wavefront and the X direction component of theelliptic phase wave.

While, in the aforementioned embodiments, the rates of variation of thespatial frequencies in the scanning direction and the cross scanningdirection are configured to be the same so that the beam radiuses in thescanning direction and the cross scanning direction measured on theimage formation surface are controlled to be the same by the fixedhologram plate, a scanning linearity can be maintained in the scanningdirection even under the condition of small optical path lengths, bymaking the rate of variation of the spatial frequency in the scanningdirection meet the same conditions as above, and by controlling thephase of the object wave so that it becomes an elliptical wave.

Since the amount of beam aberration in the scanning direction and thecross scanning direction becomes different by providing the aboveconfigurations, the variation of the amount of beam aberration iscorrected by configuring the light incident on the rotatable hologram702 so that the focal distances in the scanning direction and the crossscanning direction are different.

Another thing to be noted is that, in order to decrease the distortion,the focal distance F needs to be made larger, as is known in a convexlens optical system. Similar results can be achieved, however, by ahologram, by enlarging the distance between the hologram face and thepoint light source of the reference wave.

Enlarging the focal distance F of the reference spherical wave causesthe image formation position in the center to be further away, thusdecreasing the distortion but not cancelling it completely.

Complete cancellation of the distortion can be achieved by anon-spherical lens where F is larger nearer the lens rim, or by ahologram where the constructing wave and the reconstructing wave arecontrolled as appropriate to achieve the same effect.

This invention realizes an elliptical phase by providing, in the phaseequation of the object wave, different coefficients of X and Y for C₁and C₂ respectively, thus reducing the rate of variation of the spatialfrequency in the scanning direction, preventing an over-correction atthe scanning center, and realizing a linear scanning. This inventionalso corrects the variation in the amount of beam aberration by aconfiguration such that the light incident on the rotatable hologram 702has different focal distances in the scanning direction and the crossscanning direction. This invention succeeds in completely cancelling thedistortion by ensuring that the wave number of a spherical wave and awave number derived from the wavelength of a reconstructing wave are ofdifferent phases by virtue of coefficients a and b in the phase equationof the reference wave, so that F becomes larger toward the scanning endthan towards the scanning center.

This invention also prevents reduction of light power on the imageformation surface 704 and improves the diffraction efficiency byintroducing the term C₀ ·Y so that the fixed hologram plate 3 isprovided with an off axis characteristic where the diffracted light isdiffracted in the cross scanning direction.

Because the interference fringe distribution of the fixed hologram plate703 is manufactured by using waves, one of which has a sphericalaberration, the amount of aberration being measured along the Ydirection, astigmatism, and a coma; and the other of which waves has aspherical aberration, the amount of aberration being measured along theX direction, this other wave having a wavelength different from thewavelength of the reconstructing wave, and having astigmatism, a fixedhologram plate 703 capable of linear scanning and correction ofdistortion can easily be manufactured by exposure.

Furthermore, because the interference fringe distribution of the fixedhologram plate 703 is manufactured by using waves, one of which waveshas a spherical aberration coming from a point-light source, the wavebeing an origin of the Y direction component of an elliptical phase waveand having coma, and the other of which waves is an origin of aline-light source wavefront and the X direction component of anelliptical phase wave, a fixed hologram plate 703 capable of linearscanning and correction of distortion can be manufactured with a simpleexposure optical system.

FIG. 61 is a diagram illustrating a configuration of an embodiment ofthis invention (cross scanning direction). FIG. 62 is a diagramillustrating a configuration of an embodiment of this invention(scanning direction). FIG. 63 is a diagram describing the fixed hologramplate of an embodiment of this invention (object wave). FIG. 64 is adiagram describing the fixed hologram plate of an embodiment of thisinvention (reference wave). FIG. 65 is a diagram describing the fixedhologram plate of an embodiment of this invention (reference wave).

A diverging light from a semiconductor laser 710 is first turned into aplane wave by a collimating lens 711 in the cross scanning direction ofFIG. 61, and is then made to converge on a hologram disk 702 in thecross scanning direction Y by a Y-side cylindrical lens 713 via anX-side cylindrical lens 712. The light is diffracted by the hologramdisk 702, is then incident on the fixed hologram 703, and it isdiffracted and made to converge on the image formation surface 704a ofthe photoconductive drum 704. A diverging light from a semiconductorlaser 710 is turned into plane wave by a collimating lens 711 in thescanning direction of FIG. 62, is then converged by a X-side cylindricallens 712 hologram disk 702, and is made to converge at a point M₁ beyondthe hologram disk 702 in the scanning direction X via the Y-sidecylindrical lens 713. The light is then incident on the fixed hologramplate 703, and is diffracted and made to converge on the image formationsurface 704a on the photoconductive drum 704.

The fixed hologram plate 703 in this embodiment is configured so thatthe rates of variation of the spatial frequency in the scanningdirection and the cross direction thereof are different.

While the object wave for manufacturing the interference fringedistribution of the fixed hologram plate 703 is an isotropic sphericalwave, this invention allows the amount of the aberration in the scanningdirection and the cross scanning direction to be different from eachother, as shown in FIGS. 63(A) and 63(B). A linear scanning is achievedby providing aberration, such as creating regular distances from theoptical axis to the hologram face at any point in each scanningdirection.

This linearity is possible because a large scanning width on thehologram disk 702 under a small optical path from the hologram disk 702to the fixed hologram plate 703 causes degradation of linearity becausethe high spatial frequency of the fixed hologram plate 703 acts to drawthe diffracted light back to the scanning center.

In order to lower the spatial frequency, the distance F1 from theoptical axis to the hologram face is configured to be uniform at anypoint used for determining the amount of aberration in the scanningdirection in FIG. 63(B). The distance F1 is also configured to be longerthan the distance F2, F2 being the corresponding distance in the crossscanning direction in FIG. 63(A).

That is, in the cross scanning direction, the distance from the opticalaxis to the hologram face fulfills the equation (93) below:

    P.sub.1 Q.sub.1 =P.sub.2 Q.sub.2 =P.sub.3 Q.sub.3 . . . =F2(93)

In the scanning direction, the distance from the optical axis to thehologram face fulfills the equation (94) below:

    P.sub.1' Q.sub.1' =P.sub.2' Q.sub.2' =P.sub.3' Q.sub.3'  . . . =F1(94)

where the distances F2 and F1 fulfill the inequality (95) below.

    F1>F2                                                      (95)

In this way, the spatial frequency of the fixed hologram plate 703 inthe scanning direction will be relatively low and an over-correction oflinearity can be prevented.

The phase Φ_(O) (X, Y) of the object wave fulfilling the above equationsand recorded on the fixed hologram plate 703 is represented by theequation (96) below:

    Φ.sub.0 =k.sub.2 ·(C.sub.1 ·X.sup.2 +C.sub.2 ·Y.sup.2 +C.sub.0 ·Y)                   (96)

where ##EQU28## and where X and Y are coordinates in the scanningdirection and the cross scanning direction are respectively determinedwith respect to the origin at the scanning center of the fixed hologramplate 703, and k₂ is a wave number derived from the wavelength λ₂ of thereconstructing wave.

As can be seen from the equation (94), since C₀ ·Y is an off-axis termprovided so that the diffracted light from the fixed hologram plate 703is bent in the cross scanning direction so as to improve the diffractionefficiency, the object wave is of an elliptical phase with respect to Xand Y, as shown in FIG. 63(D), and not a circle phase derived from theisotropic spherical wave of the conventional technology as shown in FIG.63(C).

Accordingly, since the spatial frequency f_(x) in the scanning directionand the spatial frequency f_(y) in the cross scanning direction areobtained by a partial differentiation of the equation (96), they aregiven by the following equations (97) and (98). ##EQU29##

It is known from comparison of the equations (96) and (97), that thespatial frequency f_(x) can be made relatively small because it followsfrom the condition specified in the equation (95) that the rate ofvariation of the spatial frequency f_(x) in the scanning direction issmaller than the rate of variation of the spatial frequency f_(y) in thecross scanning direction.

Providing the distance F1 in the scanning direction separately from thecross scanning direction causes the beam radius on the image formationsurface 704a to be different in the scanning direction and the crossscanning direction. Correction by such provision is essential in thisembodiment.

As shown in FIG. 62, assuming that, in the cross scanning direction, animage is formed at an image formation distance L when the fixed hologramplate 703 is of a parameter F2, the focal distance d₁ of the divergingwave outgoing from the hologram disk 702 is determined from the equation(99) below, derived from a known image-formation equation, whichequation determines the focal distance required for an image formationin the scanning direction at the image formation distance L and at aparameter F1.

    1/d.sub.1 =1/F1-1/L                                        (99)

Since the position of the hologram disk 702 cannot be changed, due to arequirement from the image formation condition in the cross scanningdirection, an incident focal distance d₀ of an incident wave isadjusted.

That is, the focal distance d₀ of the incident light is adjusted by theX-side cylindrical lens 712, and the incident convergence point M₁ isset beyond the hologram disk 702. This arrangement ensures that theoutgoing wave originating position M₂ of the hologram disk 702 isremoved from the fixed hologram plate 703 by the distance d₁, so that animage is formed in the scanning direction at the image formationdistance L and at a parameter F1. This way, the beam radius variation inthe scanning direction and the cross direction can be corrected.

While in the equation (96), the wave number derived from the wavelengthλ₂ of the reconstructing wave is represented by the constant k₂ of thephase of the object wave, this wave number can be different from thewave number of the reconstructing wave.

In case the semiconductor laser 710 is used for producing areconstructing wave, the wavelength λ₂ is 780 nm. Since the wavelengthsensitivity of a hologram material having a high diffraction efficiencybelongs to a wavelength range shorter than this wavelength, manufactureof a hologram should be done with a laser light source having a shorterwavelength. For example, an argon laser having a wavelength λ₁ of 488 nmmay be used.

Thus, with the wavelength of the object wave being λ₁ and the wavelengthof the reconstructing wave being λ₂, it is required to change the phaseof the object wave.

Given that the wave number derived from the constructing wave having awavelength of λ₁ is k₁, and that the wavelength ratio between theconstructing wave and reconstructing wave is S (=λ₂ /λ₁), the phase Φ₀(X, Y) of the object wave is given by the equation (100) below.##EQU30##

Accordingly, when using a wave number k₁ derived from the wavelength λ₁of a hologram constructing wave, which wavelength is different from thewavelength of the semiconductor laser 710, the distances F1 and F2 ofthe object wave are determined by multiplying the wavelength ratio timesF1 and F2 determined as appropriate on the basis of the equation (94).

Reduction of distortion is achieved by shifting the point-light sourceposition of the spherical wave used as the reference wave, from D₀ toD_(0'), as shown in FIG. 64. Such shifting places the image formationposition at the scanning center further away, and therefore cannotensure a cancellation of distortion.

The phase Φ_(R') (X, Y) of the reference wave given by theaforementioned equation (62) is represented by the equation (101) below.##EQU31##

The phase of the reference wave is expressed by the difference betweenthe phase of the spherical wave shown by the first term of the equation(101) and the phase of the cylindrical wave shown by the second termthereof, k₂ being the wave number (2π/λ₂) derived from the wavelength ofthe reconstructing wave and being of the same value as the wave numberk₁ derived from the wavelength λ₁ of the constructing wave. Thereference wave contains a wavefront having a spherical wave optical axisD₀ O, which is the same as the distance from the hologram optical axisto the face of the hologram disk 2.

Cancellation of distortion requires that a non-spherical lens having afocal distance F(D₀ O) at the center, and an increasingly greater focaldistance toward the end is manufactured with a hologram, wherein thephase Φ_(R) (X, Y) of the reference wave for constructing the hologramshould fulfill the equation (102) below. ##EQU32##

The difference between the equations (101) and (102) is that the wavenumber k₁ derived from the wavelength λ₁ of the constructing wave isused in the equation 102 instead of the wave number k₂, used in theequation 101, derived from the wavelength λ₂ of the reconstructing wave.Increasing the wavelength λ₁ of the constructing wave to be greater thanλ₂ makes the wave number k₁ smaller than k₂, thus making the phase ofthe reference wave smaller.

Accordingly, a wave front, different from the wavefront of thediffracted wave incident on the fixed hologram plate 703 from thehologram disk 702, is recorded on the fixed hologram plate 703.Therefore, use of a reconstructing wave of wavelength λ₂ at the time ofreconstructing ensures that a non-spherical lens having a focal distanceF(D₀ O) at the center, and an increasingly greater focal distance towardthe end can be manufactured with a hologram.

This arrangement maintains the scanning center convergence at the sameposition, as shown in FIG. 65(A), thus enabling the obtaining of adistortion-free scanning beam as the scanning end convergence positionshifts toward the image formation surface 704a. As shown in FIG. 65(B),because the scanning beam shifts outward at the scanning end, a linearscanning is realized.

Although, in the phase equation (102), the coefficient k₁ is valid forthe cylindrical wave term, too, only a spherical wave can be responsiveto distortion correction. Therefore, the same effect is achieved as withthe equation (102) by using the phase equation (103), where thecoefficient k₂ for the cylindrical wave term is retained. ##EQU33##

Thus, the phase distribution Φ_(H) (X, Y) of the fixed hologram plate703, which distribution serves the purpose of linearity correction anddistortion correction, is the difference obtained when subtracting thereference wave phase Φ_(R) from the object wave phase Φ_(O), and isgiven by the equation (104) or the equation (105). ##EQU34##

FIG. 66 is a spot diagram of an embodiment of this invention.

In a light beam scanning apparatus shown in FIGS. 61 and 62 employingthe fixed hologram plate 703 that has a phase distribution determinedfrom the equation (105), different beam radiuses shown in FIG. 66 areobtained at each position in the range from the scanning center to thescanning end. The beam radius is maintained smaller than 60 microns,small enough to be applied to 400 dpi resolution.

A linearity of below a 0.7% level and deviation from a straight line, ofthe photoconductive drum 704, below a 200 micron level result from thisembodiment. Other benefits include little scanning-position variationdue to wavelength variation of the semiconductor laser.

In this way, a linear scanning and a light scanning free from distortioncan be achieved even when the distance between the hologram disk 702 andthe image formation surface 704a is maintained small for purposes ofdownscaling. (See a description of an embodiment of manufacture of thefixed hologram plate 703.)

FIGS. 67 and 68 are diagrams (Part 1 and Part 2) illustrating amanufacture of the fixed hologram plate of an embodiment of thisinvention. Since the phase function Φ_(H) (X, Y) of the fixed hologramplate 703 is represented by the above equation (105), the spatialfrequency (f_(x) in the direction X, and f_(y) in the direction Y) ofthe fixed hologram plate 703 is obtained by carrying out a partialdifferentiation on the equation (105), the result thereof being theequations (106) and (107) below. ##EQU35##

A hologram on which the above wavefront is recorded can be manufacturedby an electron beam or a laser drawing, but the manufacture takes timesince the area concerned is large. The use of a hologram as an auxiliaryoptical system in an exposure invites a degradation of S/N ratio.

Therefore, an exposure optical system is needed for manufacturing ahologram having a high S/N ratio, which optical system generates thewavefront described above.

It is also to be noted that a wave having a wavelength different fromthat of the reconstructing wave may be needed in the manufacture, inconsideration of the sensitivity of the hologram material.

Examining the equations (106) and (107), 2C₁ ·X as well as 2C₂ ·Ysignifies the beam in FIG. 67(A), and indicates a wave of an ellipticalphase, as shown in FIG. 63.

The third term in each of the equations (106) and (107), with bexcluded, signifies a beam from the point-light source having thecoordinates (0, Y₀, Z₀), as shown in FIG. 67(B). With b included, thethird term in each equation signifies waves having spherical aberration.This is equivalent to the hologram lens reconstructed with thewavelength altered.

The second term of the equation (106) signifies a beam from theline-light source passing the coordinate (0, 0, Z₀), as shown in FIG.67(C), while the second term C₀ of the equation (107) signifies adiagonally incident plane wave, as shown in FIG. 68(A), a coma beingcreated by combining this beam with other beams.

The above arrangement is the same as allowing a beam to go diagonallyinto a lens, as shown in FIG. 68(B).

Accordingly, the wave obtained by this arrangement is a wave having, inthe direction X, the wavefront from the line-light source of FIG. 67(C),the spherical aberration from the point-light source of FIG. 67(B), andthe X component of the wave of FIG. 67(A) having an elliptical phase andhaving, in the direction Y, the spherical aberration from the pointlight source of FIG. 67(B), the coma of FIG. 68(A), and the Y componentof the wave of FIG. 67(A) having an elliptical phase.

That is, two waves are constructed, namely, one wave having a wavefrontfrom the line-light source, a spherical aberration from the point-lightsource, and an X component of the wave having an elliptical phase; and awave, having a spherical aberration from the point-light source, a coma,and a Y component of the wave having an elliptical phase. Aninterference fringe distribution having the phase function of theequation (105) is obtained by the interference of these two waves.

A construction of these wavefronts is achieved as follows: a sphericalaberration is constructed by allowing a beam to be incident on aplano-convex lens in an on-axis manner, as shown in FIG. 68(C); a comais constructed either by allowing a beam to be diagonally incident on aplano-convex (or plano-concave) cylindrical lens, as shown in FIG.68(D), or by allowing a beam to be incident on a plano-convex (orplano-concave) cylindrical lens in an off-axis manner, as shown in FIG.68(E).

A wave having an elliptical phase is constructed with the use ofplano-concave (or plano-convex) cylindrical lenses disposed at rightangles to each other, that is one in the direction X and the other inthe direction Y, as shown in FIG. 68(F).

Since the X's in the numerator and the denominator of the equation (106)relative to the direction X cancel each other, the rate of variation isrelatively small. The (Y-Y₀)'s in the numerator and the denominator ofthe equation (107) relative to the direction Y also cancel each other,but the denominator thereof contains X of a great variation rate, and sois subject to a greater change.

Therefore, a hologram construction optical system is designed such thattwo lenses are provided separately, namely a lens functioning in thelongitudinal direction X of the fixed hologram plate 703, and a lensfunctioning in the direction Y, which is at right angles to thedirection X. A relatively simple optical system is provided in thedirection X, in which direction the rate of variation is small.

Accordingly, the line-light source wavefront (an origin of sphericalaberration, and astigmatism) and the X component of the wave having anelliptical phase are obtained when the reference-wave side is designatedas the direction X; the spherical aberration coming from the point-lightsource, the coma, and the Y component of the wave having an ellipticalphase are achieved when the object wave side is designated as thedirection Y.

Although spherical aberration is effective in both the reference andobject sides, it is utilized in the object side. FIG. 69 describes thefirst embodiment of the exposure system for the manufacture of ahologram of this invention. Specifically, FIG. 69(A) is a diagramillustrating the configuration thereof as viewed from the face formed bythe coordinate axes Y and Z, while FIG. 69(B) is a diagram illustratingthe configuration as viewed from the face formed by the coordinates axesX and Z.

The X component of the wave having an elliptical phase, which componentconstitutes the reference wave, and the line-light source wavefront ofthe reference wave are generated by allowing a diverging wave from thepoint-light source 750 to pass through the X-side plano-convexcylindrical lens 751, and allowing it to be incident on the fixedhologram plate 703.

The spherical aberration of the object wave is created by allowing adiverging wave from the point light source 760 to pass through theplano-convex lens 761. The Y component of the object wave has anelliptical phase, and the coma of the object wave are created byallowing the outgoing wave, outgoing from the plano-convex lens 761, topass through the plano concave lens 762 on the Y side before allowing itto be incident on the fixed hologram plate 703.

In accordance with this configuration, the line-light source wavefront(an origin of spherical aberration and astigmatism) and the X componentof the wave having an elliptical phase are created with thereference-wave side designated as the direction X. The sphericalaberration and coma coming from the point-light source, and the Ycomponent of the wave having an elliptical phase are created with theobject-wave side designated as the direction Y, thereby allowing theobtaining of the phase distribution of the equation (105).

FIG. 70 is a diagram describing the second embodiment of the exposuresystem for the manufacture of a hologram of this invention, and showinga configuration thereof as viewed from the face formed by the coordinateaxes Y and Z, wherein the parts that are the same as parts shown in FIG.69 are given the same reference numerals as in the previous figure.

In this embodiment, a plano-convex cylindrical lens 763 is added to theobject wave side.

This is because, although the configuration of FIG. 69 is applicable toa case where the object-wave side and the reference-wave side areseparated enough to allow insertion of each lens therebetween, theconfiguration is not applicable to a case where, in the object-waveside, the image formation surface blocks the reference-wave side. Theplano-convex cylindrical lens 762 alone cannot assure a sufficientamount of coma.

Therefore, when creating the object wave, a diverging wave from thepoint-light source 760 is allowed, in order to generate the sphericalaberration, to pass through the plano-convex lens 761, the wave is thenallowed, in order to generate the coma, to pass through the Y-sideplano-convex cylindrical lens 763, and is allowed, in order to generatethe Y-component of the wave having an elliptical phase and the coma, topass through the Y-side plano-concave cylindrical lens 762, before beingallowed to be incident on the fixed hologram plate 703.

This arrangement ensures that, even when the Y-side length of the fixedhologram plate 703 is great, a lens can be inserted so that the coma isgreater than otherwise.

FIG. 71 shows the third embodiment of the exposure system for themanufacture of a hologram of this invention. Specifically, FIG. 71(A) isa diagram showing a configuration thereof as viewed from the face formedby the coordinate axes X and Z, and FIG. 71(B) is a diagram showing theconfiguration as viewed from the face formed by the coordinate axes Yand Z.

In this embodiment, two plano-convex cylindrical lenses 764 and 765 areprovided on the object-wave side, in addition to those already providedin the configuration shown in FIG. 70, so that a large enough coma isobtained.

The X component of the wave having an elliptical phase, which componentconstitutes the reference wave, and the line-light source wavefront ofthe reference wave are generated by allowing a diverging wave from thepoint light source 750 to pass through the X-side plano-convexcylindrical lens 751 before allowing it to be incident on the fixedhologram plate 703.

Therefore, when creating the object wave, a diverging wave from thepoint-light source 760 is allowed, in order to generate the sphericalaberration, to pass through the spherical plano-convex lens 761, thewave is then allowed, in order to obtain the necessary coma, to beoff-axially incident on three Y-side plano-convex cylindrical lenses764, 765, and 763, and is allowed, in order to generate the Y-componentof the wave having an elliptical phase and the coma, to pass through theY-side plano-concave cylindrical lens 762 before being allowed to beincident on the fixed hologram plate 703.

In accordance with this configuration, the line-light source wavefront(an origin of spherical aberration and astigmatism) and the X componentof the wave having an elliptical phase are created with thereference-wave side designated as the direction X. The sphericalaberration and coma coming from the point light source, and the Ycomponent of the wave having an elliptical phase are created with theobject-wave side designated as the direction Y, thereby allowing theobtaining of the phase distribution of the equation (105). Thus, many ofthe lenses in the exposure system share a common optical axis andtherefore allow for easy adjustment.

The fixed hologram plate 703 thus manufactured is used as the original,from which a mold is taken. The mold is then used to reproduce a fixedhologram plate 703 of the same characteristics.

In addition to the above-described embodiments, this invention can beextended to variations such as follows.

In a manufacture of the fixed hologram plate in this invention, it ispossible to employ a method (a so-called electron-beam exposure method)where an electron beam is controlled by a computer in accordance withthe above equations so that predetermined interference fringes areformed on a hologram dry plate. Another method (a so-called laser-beamexposure method) can also be employed, as described before, whereby alaser beam is controlled by a computer in accordance with the aboveequations.

These methods are suitable for cases where the fixed hologram plate ismass-produced, or where the size thereof or the deflection mode of thelight is changed from time to time. While these methods include computercontrol and have a disadvantage of leading to large size of apparatusesfor such manufacture, the above-mentioned embodiments allow a relativelyinexpensive manufacture of the fixed hologram plate with a smallapparatus for the manufacture.

Although the above description was given assuming that the light-beamscanning apparatus is applied in an electrophotograph printingapparatus, it could also be applied to a laser drawing apparatus or to alaser inspection apparatus.

Although a disk shaped rotatable hologram was assumed in thedescription, a drum shaped rotatable hologram can also be employed.

As has been described, an elliptical phase is realized in this inventionby providing, in the phase equation of the object wave, differentcoefficients X and Y with respect to C₁ and C₂. This invention alsoenables the lowering of the rate of variation of the spatial frequencyin the scanning direction, and the preventing of over-correction of thescanning center, thus realizing a linear scanning. Variations in theamount of beam aberration can be corrected by configuring so that thelight incident on the rotatable hologram 702 has different focaldistances in the scanning direction and the cross scanning direction.Further, F of the scanning beam in this invention is configured to belarger toward the scanning end than at the scanning center, so thatdistortion is cancelled by ensuring that the wave number of thespherical wave is different from the wave number derived from thewavelength of the reconstructing wave by adjusting the coefficients aand b in the phase equation of the reference wave.

This invention is not limited to the embodiments described, butvariations and modifications are possible without departing from thescope of this invention.

FIG. 72 illustrates a facet hologram of the rotatable hologram used inthe aforementioned first invention. As shown in the figure, a planegrating hologram, in which hologram the interference fringe distributionis uniform as far as the positions of the fringes are concerned, is usedin a facet hologram 820a of a rotatable hologram 802a used in the firstembodiment of the present application.

In the figure, it is assumed that a laser-light incident on the points Aand B on the rotatable hologram 802a of FIG. 72(A) passes through thepoints A' and B' on a fixed hologram plate 803a. The spatial frequencymeasured from the rotational center of the rotatable hologram 802atoward the scanning center thereof is the same at the points A and B onthe rotatable hologram 802a, and the spatial frequency is zero in thedirection X, which is at right angles to that direction.

Accordingly, the spatial frequency measured at the position on which alaser light is incident does not exhibit a change even when therotatable hologram 802a rotates. When a laser light incidence angle intothe rotatable hologram 802a, the angle being measured at the scanningcenter, and the outgoing angle thereof are determined,the laser lightscanning track on the fixed hologram plate 803a is determined.

However, recent demand for a downscaled laser printer, for example,requires that a light-beam scanning apparatus be made smaller. Toachieve this, the optical path length from the rotatable hologram 802ato the image formation surface needs to be small. It should be notedthat shortening the optical path from the rotatable hologram 802a to thefixed hologram plate 803a causes the scanning width to be small.

On the other hand, shortening the optical path from the fixed hologramplate 803a to the image formation surface causes the fixed hologramplate 803a not to perform the optical functions desired of it, forexample, the functions including a linearity correction function, astraight-line scanning function, and a laser light image formationfunction.

Accordingly, the objects of this invention are to shorten the opticalpath from the rotatable hologram to the image formation surface whilemaintaining the optical characteristics of the fixed hologram plate, sothat the downscaling of an apparatus may be achieved.

FIG. 73 is a diagram showing a configuration of an embodiment of thisinvention.

As shown in FIG. 73(B), a facet 820 of a rotatable hologram 802 enablesthe changing of the spatial frequency, at the position on which a laserlight is incident, when a hologram is used where a change indistribution is seen only in the spatial frequency measured from therotation center M of the rotatable hologram 802, in the direction Y,toward the scanning center position on the rotatable hologram 802.

That is, calling the position on the rotatable hologram 802, on whichposition the laser light is incident, r₀, calling the spatial frequencyat the position r (the distance thereto being measured from the rotationcenter M of the rotatable hologram 2, in the direction Y, toward thescanning center position on the rotatable hologram 802) on the rotatablehologram 802, f, and calling the spatial frequency (the spatialfrequency at the position r₀) determined from the incident and outgoingangle of the laser light incident on the rotational center M of therotatable hologram 802, f₀, the spatial frequency distribution measuredfrom the rotational center of the rotatable hologram 802 in thedirection of the scanning center position on the rotatable hologram 802is set to be of a value as determined from the equation (121) below.

    f=f.sub.0 +a×(r-r.sub.0)                             (121)

where a, the rate of variation of the spatial frequency, is a constant.

Assuming that a is negative, the spatial frequency f at the position rbecomes higher than the spatial frequency f₀ at the position r₀. Thespatial frequency will then be of a distribution where the frequency ishigher toward the center of the rotatable hologram 802, with the resultthat the diffraction angle in the scanning direction can be large andthe scanning width can be large.

With this arrangement, it is possible to shorten the optical path fromthe rotatable hologram 802 to the fixed hologram plate 803 withoutchanging the scanning width on the image formation surface.

If the optical path from the fixed hologram 803 to the image formationsurface 804 is maintained constant, the convergence characteristics andthe deviation from a straight line of the scanning on the imageformation surface 804 deteriorate to the extent that the optical pathfrom the rotatable hologram 802 to the fixed hologram plate 803 isshortened. Therefore, the optical path from the fixed hologram plate 803to the image formation surface 804 is controlled to be slightly longer.

As for the fixed hologram plate 803, the direction cosine 1 in thedirection X and the direction cosine m in the direction Y are given bythe equations (122) and (123), below, where the scanning direction isrepresented by X, the cross scanning direction by Y, and the directionat right angles to these, by Z. The wavefront expressed by theseequations is realized by the fixed hologram plate 803; ##EQU36## wherea, b, c₀, d, e, y₀, and z₀ are constants.

FIG. 74 shows the manufacture of a rotatable hologram of an embodimentof this invention. As shown in FIG. 74, the rotatable hologram 802 ismanufactured using the interference of the object wave and the referencewave.

For a manufacture of a plain grating disk, the object wave and thereference wave should be plane waves. In order to provide spatialfrequency variation toward the rotational center, the intervals of theplane wave constituting the object wave should be varied toward therotational center.

To achieve the above in producing an object wave, a plane wave isprovided with a divergence property by means of a cylindricalconcave-lens 805, and is allowed to be incident on the rotatablehologram 802.

In this way, the angle formed by the wave emitted from the cylindricalconcave lens 805 and the plane reference wave is greater toward therotational center, i.e., θr>θr₀, thus ensuring a greater density and agreater spatial frequency where appropriate.

In practical terms, since the light power of the semiconductor laser,which is the light source of the apparatus, is not sufficient for themanufacture of a hologram, a light source whose light power is greater,such as an argon laser, is utilized.

The wavelength λ₀ the semiconductor laser is 780 nm, while thewavelength λ₁ of the argon laser is 488 nm, meaning that differentwavelengths are used at the construction time and at the reconstructiontime.

Therefore, as shown in FIG. 74(B), it is required that an irradiationangle λ₂ of the object wave and an irradiation angle λ₁ of the referencewave be set, at the wavelength λ₀ of the semiconductor laser, inaccordance with the equation (124) below, so that the spatial frequencyf is obtained.

    f=(sin θ.sub.1 + sin θ.sub.2)/λ.sub.0   (124)

Thereafter, as shown in FIG. 74(C), the irradiation angle θ_(2') of theobject wave and the irradiation angle θ_(1') of the reference wave areobtained from the equation (125) below, which angles are determined inorder to produce the spatial frequency f on the basis of the argon laserwavelength of λ₁.

    f=(sin θ.sub.1 '+ sin θ.sub.2 ')/1             (125)

If, as shown in FIG. 74(A), the plane object wave and the waveirradiated from the convex lens 805 are irradiated on the rotatablehologram 802 at the wavelength λ₁ using the irradiation angle θ_(2') ofthe object wave and the irradiation angle θ_(1') of the reference wave,a hologram having a predetermined spatial frequency can be manufacturedby means of a semiconductor laser having a wavelength of λ₀.

FIG. 75 is a spot diagram taken in an embodiment of this invention.

While in a conventional plane grating disk, the distance L1 between therotatable hologram 802 and the fixed hologram plate 803 is 295 mm andthe distance L2 between the fixed hologram plate 803 and the imageformation surface 804a is 255 mm, the use of the rotatable hologram 2having the spatial frequency distribution as described in FIG. 73 servesto shorten the overall optical path by enabling: L1=225 mm and L2=265mm.

The convergence characteristics and deviation from a straight line onthe image formation surface 804 is maintained by slightly increasing thedistance L2 between the fixed hologram plate 803 and the image formationsurface 804a.

The beam diameters in this arrangement measured in the range from thescanning center to the scanning end are shown in FIG. 75, where themaximum diameter is 60 microns at the scanning end. A 400 dpi resolutionis thus guaranteed.

A linearity of below a 0.5% level and a deviation from a straight line,on the drum 804, of below 100 microns are obtained. It is found that thescanning position variation due to a wavelength variation of thesemiconductor laser is extremely small, thus making the embodiment fullypracticable.

Thus, it is possible with this embodiment to enlarge the scanning widthof the rotatable hologram 802 so that the distance between the rotatablehologram 802 and the fixed hologram plate 803 can be made small, and sothat a downscaling of the apparatus can be achieved, while at the sametime it is possible to prevent variation of the scanning beam positiondue to a wavelength variation (variation of the center frequency, or ashift in multi-mode distribution) accompanying a temperature variationof the semiconductor laser, and to prevent a decrease in linearity,deviation from a straight line, and beam radius.

As has been explained in the description of this invention, the opticalpath from the rotatable hologram 802 to the fixed hologram plate 803 canbe shortened by a configuration such that the spatial frequency isgradually higher toward the edge of the beam irradiation position thanat the central beam irradiation position on the hologram 820 of thefacet of the rotatable hologram 802, so that the diffraction angle inthe scanning direction is enlarged and the scanning width is enlarged. Adownscaling is achieved in this configuration because the scanning widthis maintained constant.

While it was assumed in the above-mentioned embodiment that the beamscanning apparatus is applied to an electrophotographic printingapparatus, it can also be applied to a laser drawing apparatus or to alaser inspection apparatus. While the hologram interference fringes inthe above-mentioned embodiment were assumed to be of a parallel linepattern, it could be of other shapes. This invention is not limited tothe above embodiments but variations and modifications are possiblewithout departing from the scope of this invention.

This invention relates to possible shapes of a laser-light spot from alaser light incident on a rotatable hologram. FIG. 76 is an enlargeddiagram illustrating a facet of a rotatable hologram 902a. Also, thefigure schematically shows that the beam incidence position changes asthe rotatable hologram 902a rotates (the change being in the directionA→B→C).

As shown in the figure, the laser light beam incident on the rotatablehologram 902a is of an elliptical shape in cross section. The ellipticalshape in the light beam scanning apparatuses described so far is suchthat the minor axis of the ellipsis is along the radius of the rotatablehologram 902a.

Assuming that the angle formed by the positions A and B, which positionsare both hit by a beam, the central line passing through the rotationalcenter is θ_(b), and that the range in which a hologram is formed by themovement of the laser light beam from the position A to the position Bis θ_(h), the ratio of the angles θ_(b) and θ_(h) is defined as thevalid hologram ratio H (H=θ_(b) /θ_(h)).

In order to utilize a rotatable hologram 902a effectively, it isdesirable that this valid hologram ratio is as close to 1 as possible.

However, in the light beam scanning method shown in FIG. 76, since themajor axis of the incident beam section (the beam having an ellipticalshape) and the rotation direction of the rotatable hologram 902a arealigned, there is a certain inherent limit to the approximation of thevalid hologram ratio H to 1. That is, the rotatable hologram 902a couldfail to function efficiently because of the unused portion θ_(D) (=θ_(h)-θ_(b)) in a facet of the rotatable hologram 902a, which unused portionis inevitably created due to the fact that the beam shape is an ellipsewhose major axis lies in the rotational direction. This failure givesrise to a problem in that the scanning efficiency is lowered.

One approach to solving this problem is to turn the beam shape of thelaser light into a circle. However, the above-mentioned major axisshould be of a length large enough for the fixed hologram plate toconverge the scanning beam in the scanning direction and cannot besmaller than a predetermined size (that is, it cannot be circular inshape).

The object of this invention is to enable an effective use of arotatable hologram by enabling an approximation of the valid hologramratio H to 1 even when the beam shape of the laser light is elliptical.

FIG. 77 illustrates the principle of this invention. This invention ischaracterized in that the rotational direction of the rotatable hologram902 (shown by the arrow in the figure) and the minor axis of anelliptically-shaped beam 910 of the laser light irradiated by the lightsource (a semiconductor laser, for example, but not shown in the figure)are aligned.

With this configuration, it is possible to control the unused portionθ_(D') (=θ_(h') -θ_(b')) inevitably created in a facet 920 of therotatable hologram 902 to be small, as shown in the figure, and toapproximate the valid hologram ratio, H, to 1.

FIGS. 78 through 80 illustrate the embodiments of this invention.

FIG. 78 is a diagram illustrating, in magnification, a facet 920 of therotatable hologram 902 used in this invention. As described above, sincethis invention is configured such that the rotational direction of therotatable hologram 902 and the minor axis of the elliptically-shapedlaser light beam 910 are aligned, a hologram needs to be designed sothat an appropriate light beam scanning can be achieved with thisconfiguration. Accordingly, the spatial frequency distribution in thisinvention is set according to the equations below.

    f.sub.x =0

    f.sub.y =f.sub.0 +ΣC.sub.i *(r* sin θ).sup.i

where θ is the rotation angle of the rotatable hologram 902, r is theincident radius of the beam, C_(i) is the rate of variation of thespatial frequency (non-spherical coefficient), and f₀ is the spatialfrequency when the rotation angle θ is zero. A light beam scanningapparatus described below can be realized by using the rotatablehologram 902 having the above spatial frequency distribution.

FIG. 79 shows that the angle formed by the segment connecting theincidence position of the beam from the light source to the rotationalcenter of the rotatable hologram 902, with the segment connecting therotational center of the rotatable hologram and the fixed hologram plate903 is set to be 90°.

With this configuration, the rotational direction of the rotatablehologram 902 and the minor axis of the elliptically-shaped laser lightbeam 910 section can be aligned, so that the valid hologram ratio H isapproximated to zero.

FIG. 80 illustrates an example for which the valid hologram ratio H isactually computed. That an effective usage of the rotatable hologram 902becomes possible with this configuration is evident in that while in theconfiguration shown in FIG. 76 (shown in FIG. 80(A)), the valid hologramratio H is 0.844, the valid hologram ratio H is improved to become 0.933in the method of this invention shown in FIG. 80(B).

FIG. 81 shows first to tenth (i=1-10) order terms of the above-mentionedspatial frequency. As shown in the figure, it is found, on the basis ofa simulation, that the rotational direction of the rotatable hologram902 and the minor axis of the irradiated beam cross-section can bealigned by providing a spatial frequency distribution consisting of evenfunctions.

As has been demonstrated, the valid hologram ratio H can be increased byaligning the rotational direction of the rotatable hologram and theminor axis of the incident beam cross-section. The valid hologram ratioH can be increased and the light beam scanning apparatus that enablesthe effective utilization of the rotatable hologram 902 can be realizedby a configuration such that the beam position and the center of thefixed hologram plate 903 are points that lie on lines forming a 90°angle.

As has been described, the light beam scanning apparatus according tothe present invention can provide a simple and inexpensive opticalsystem by using two holograms and can realize a highly reliableapparatus free from scanning beam displacement caused by a wavelengthvariation of the semiconductor laser. Therefore, the scanning apparatusof the present invention is useful as a laser scanning optical system tobe incorporated in office automation equipment such as a laser printerand a laser facsimile, or in a laser drawing apparatus, and a laserinspection apparatus, for example.

We claim:
 1. A light-beam scanning apparatus comprising:an opticalsource producing a source optical beam; a scanning surface; a rotaryhologram disk having a central axis and an outer circumference with therotary hologram disk being rotatable about the central axis, the rotaryhologram disk being positioned between said optical source and saidscanning surface in the path of said source optical beam produced bysaid optical source,said rotary hologram disk being dividedcircumferentially around said central axis into a plurality of facetssuch that each facet occupies a portion of the outer circumference, theportion of the outer circumference occupied by each facet having amiddle point that substantially equally divides the portion of the outercircumference, each facet at least partially containing an imaginaryradial line extending from the central axis to the middle point of theportion of the outer circumference, each facet carrying a firstdiffraction grating pattern that produces a diffraction beam as a resultof diffraction of said source optical beam, said first diffractiongrating pattern including, in each facet, a plurality of straight linesextending parallel with each other in a direction generallyperpendicular to the imaginary radial line of the facet, in each of saidfacets, said plurality of straight lines being disposed with a pitchthat increases away from the central axis and toward the outercircumference such that the straight lines are further apart toward theouter circumference; a motor for rotating said rotary hologram diskabout said central axis; and a fixed hologram disposed between saidrotary hologram disk and said scanning surface in the path of saiddiffraction beam from said rotary hologram disk, said fixed hologramcarrying a second diffraction grating pattern that produces a scanningoptical beam as a result of diffraction of said diffraction beam, saidscanning surface being scanned by said scanning optical beam.
 2. Alight-beam scanning apparatus as claimed in claim 1, wherein said pitchchanges linearly, in each of said plurality of facets, with a distancein a direction connecting a center of said rotary hologram disk and acenter of said facet.
 3. A light-beam scanning apparatus as claimed inclaim 1, wherein said first diffraction pattern on said rotary hologramdisk is substantially equal, in each of said plurality of facets, to aninterference fringe pattern formed by an interference of a referenceplane wave and an object wave passed through a cylindrical lens.
 4. Alight-beam scanning apparatus comprising:an optical source producing asource optical beam; a scanning surface; a rotary hologram disk having acentral axis and an outer circumference with the rotary hologram diskbeing rotatable about the central axis, the rotary hologram disk beingpositioned between said optical source and said scanning surface in thepath of said source optical beam produced by said optical source,saidrotary hologram disk being divided circumferentially around said centralaxis into a plurality of facets such that each facet occupies a portionof the outer circumference, the portion of the outer circumferenceoccupied by each facet having a middle point that substantially equallydivides the portion of the outer circumference, each facet at leastpartially containing an imaginary radial line extending from the centralaxis to the middle point of the portion of the outer circumference, eachfacet carrying a first diffraction grating pattern that produces adiffraction beam as a result of diffraction of said source optical beam,said first diffraction grating pattern including, in each facet, aplurality of straight lines extending parallel with each other andparallel with the imaginary radial line of the facet, in each of saidplurality of facets, said plurality of straight lines being disposedwith a pitch that decreases away from the imaginary radial line suchthat the straight lines get closer together with increasing distanceaway from the imaginary radial line; a motor for rotating said rotaryhologram disk about said central axis; and a fixed hologram disposedbetween said rotary hologram disk and said scanning surface in the pathof said diffraction beam from said rotary hologram disk, said fixedhologram carrying a second diffraction grating pattern that produces ascanning optical beam as a result of diffraction of said diffractionbeam, said scanning surface being scanned by said scanning optical beam.5. A light-beam scanning apparatus as claimed in claim 4, wherein saidpitch decreases linearly, in each of said plurality of facets, with adistance from said center of said facet in a direction perpendicular tosaid direction connecting said center of said rotary hologram disk andsaid center of said facet.
 6. A light-beam scanning apparatus as claimedin claim 4, wherein said first diffraction pattern on said rotaryhologram disk is substantially equal, in each of said plurality offacets, to an interference fringe pattern formed by an interference of areference plane wave and an object wave passed through a cylindricallens.